Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 60, 1-32; http://www.math.u-szeged.hu/ejqtde/

FRACTAL ANALYSIS OF HOPF BIFURCATION FOR A CLASS OF COMPLETELY INTEGRABLE NONLINEAR SCHRÖDINGER CAUCHY

PROBLEMS

JOSIPA PINA MILIŠIĆ, DARKO ŽUBRINIĆ, AND VESNA ŽUPANOVIĆ

Abstract. We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in L^{2}(Ω) is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in ℝ^{n}.

Corresponding author: Josipa Pina Milišić University of Zagreb, Faculty of Electrical Engineering and Computing Unska 3, 10000 Zagreb, Croatia

pina.milisic@fer.hr

1. Introduction

1.1. Motivation and formulation of the problem. Dimension theory for dynamical systems has been rapidly developed due to its important applications in other natural, social and technical sciences. Since the early 1970’s scientists have started to estimate and compute fractal dimensions of strange attractors for finite (Lorenz, Henon, Chua, Leonov, etc.), as well as for the infinite-dimensional dynamical systems (Ladyzhenskaya, Foiaş and Temam, Babin and Vishik, Ruelle, Lieb, etc.) Fractal dimensions in dynamics are discussed in a survey article [29]. However, our approach to the question of fractal dimensions in dynamics is different in a way that we are interested in the complexity of trajectories of dynamical systems and study the dependence of the computed fractal dimension upon the bifurcation parameter. Knowing the information about fractal dimension of trajectories

1991 Mathematics Subject Classification. 35Q55, 37G35, 28A12, 34C15. Key words and phrases. Schrödinger equation, Hopf bifurcation, box dimension, Minkowski content,

compactness, rectifiability, bundle of trajectories, oscillation, multiple spiral, spiral chirp.

The authors have been supported by the Austrian-Croatian Project of the Austrian Exchange Service (ÖAD) and the Ministry of Science, Education, and Sports of the Republic of Croatia (MZOS).

EJQTDE, 2010 No. 60, p. 1 enables to measure the complexity of the studied dynamics which we mostly describe by calculating the corresponding box dimension and, in order to describe finer properties, its Minkowski content. Namely, the Hausdorff dimension of trajectories that we are interested in, is always trivial.

The second and third author have undertaken a systematic study of fractal properties of trajectories of vector fields near the weak focus in ℝ^{2} and analogously in ℝ^{3}. Furthermore, they considered a connection between the box dimension of the trajectory and the bifurcation of the related dynamical system. Here we mention that this interesting connection has been studied also in [10, 16] but for one-dimensional discrete dynamical systems.

In this article, using methods developed in [20, 26, 27, 28] we consider the trajectories of vector fields in infinite-dimensional case and we calculate their box dimensions. The original idea of this paper grounds on the connection between nonlocal Schrödinger evolution problems and the corresponding system of ODE’s. More precisely, starting from completely integrable nonlinear integro-differential Schrödinger equation we arrive at an equivalent system of infinitely many nonlinear ODE’s. Concerning the observed link between Schrödinger problems and vector fields it is interesting to point out that each planar system of ODE’s with polynomial right-hand side can be interpreted as a nonlinear Schroedinger equation with an explicit corresponding nonlinear term. Moreover, as it is described in Section 6.3, this consideration is valid even for any dynamical system in ℝ^{n}. Therefore, it is worth noting that the 16th Hilbert problem about the search of an uniform upper bound for the number of limit cycles in polynomial vector fields can be considered in terms of a Schrödinger equation.

In Žubrinić and Županović [26, Theorem 9] it has been shown that the box dimension of spiral trajectories of the classical Hopf bifurcation system in the plane described by (4), viewed near the focus, is equal to d = 4/3 when the bifurcation parameter a_{0} is equal 0. Furthermore, these trajectories are Minkowski nondegenerate, i.e. their d-dimensional Minkowski contents, are different from 0 and ∞. On the other hand, for a_{0} ≠ 0 all trajectories have trivial box dimension equal to 1, due to the fact that we have strong focus in this case. We shall see that it is possible to obtain analogous results in infinitedimensional case for problem (1). The main results are stated in Theorem 11 (Minkowski content), Theorem 15 (box dimension of multiple spirals), Theorem 17 (box dimension of solutions of NLS Cauchy problems), and Theorem 18 (box dimension of spiral chirps). In Theorem 11 we obtain some surprising relations between the sequence of Minkowski contents of projections of the solution of (1) at Hopf bifurcation, and the Sobolev and Lebesgue norms of the initial function u_{0}.

We start with the Cauchy problem for the following nonlinear Schrödinger (NLS) initialboundary value problem:

(1)  \{ \begin{matrix}u_{t}(t, x) & = & iΔu(t, x) − γ u(t, x) (∫_{Ω} |u(t, x)|^{2}dx + a_{0}) \\ u(t, x) & = & 0 for x ∈ ∂Ω, t ∈ (t_{min}, t_{max}) \\ u(0, x) & = & u_{0}(x) for x ∈ Ω. \end{matrix}

where i is the imaginary unit, γ is a fixed nonzero complex number, a_{0} a real bifurcation parameter, Ω a bounded domain in ℝ^{N}, N ≥ 1, and u_{0} : Ω → ℂ is a given initial function, u_{0} ∈ L^{2}(Ω, ℂ). Here Δu = ∑_{j=1}^{N} \frac{∂^{2}u}{∂x_{j}^{2}} is the Laplace operator. For a fixed space variable x, the solutions u : (t_{min}, t_{max}) → L^{2}(Ω, ℂ) of the NLS Cauchy problem (1) can be considered as trajectories in the Hilbert space L^{2}(Ω, ℂ) where L^{2}(Ω, ℂ) = L^{2}(Ω) + iL^{2}(Ω) is the complexification of the real space L^{2}(Ω). We point out that t_{min} and t_{max} depend on γ, a_{0} and u_{0}, and 0 ∈ (t_{min}, t_{max}). In this article, we are dealing with unbounded time intervals (t_{min}, t_{max}) either of the form (t_{min}, ∞), or (−∞, ∞), or (−∞, t_{max}).

Due to the integral term in (1), this NLS Cauchy problem is of nonlocal type. Similar nonlocal problems with the same integral term have been considered for the modified cubic wave equation,

u_{tt} − Δu + c (∫_{Ω} u(t, x)^{2}dx) u = 0,

with homogeneous Dirichlet boundary condition, in Cazenave, Haraux and Weissler [3, 4, 5], in their study of completely integrable abstract wave equations. We consider the Schrödinger equation (1) for a_{0} = 0 as an approximation of the equation u_{t} = iΔu−γu|u|^{2} following the approach of [5, p. 130].

The same integral term as in (1) can be seen in Christ [8, pp. 132 and 133]. An analogous one appears in an equation of fourth order arising from the theory of aeroelasticity,

u_{xxxx} + (α − β ∫_{0}^{1} u_{x}^{2}dx) u_{xx} + γu_{x} + δu_{t} + εu_{tt} = 0

see Chicone [7, p. 310]. Other integro-differential Schrödinger problems involving different integral operators on the right-hand side have been studied in numerous papers, see for example Chen and Guo [6].

1.2. Interpretation in ℓ_{2}. In order to write down problem (1) as an infinite system of nonlinear ordinary differential equations we exploit a well known Fourier series expansion using the decomposition with respect to a Hilbert basis in L^{2}(Ω). Let (φ_{j}) be the orthonormal basis of eigenfunctions of the operator −Δ with zero boundary data and with eigenvalues (λ_{j}) such that 0 < λ_{1} < λ_{2} ≤ λ_{3} ≤ … (see e.g. Brezis [1, ersatz (12) on p. 209]). The basis (φ_{j}) is contained in the real Lebesgue space L^{2}(Ω), and therefore it is also

EJQTDE, 2010 No. 60, p. 3 an orthonormal base in L^{2}(Ω, ℂ). By writing the solution u(t, x) in the form:

(2)  u(t, x) = ∑_{j=1}^{∞} z_{j}(t)φ_{j}(x),

the NLS Cauchy problem (1) formally reduces to a lattice Schrödinger equation on the Hilbert space of quadratically summable sequences of complex numbers, which we denote by ℓ_{2}(ℂ). More precisely, we obtain an infinite system of nonlinear ODE’s:

(3)  ż_{j} = −iλ_{j}z_{j} − γz_{j}(∥z∥^{2} + a_{0}), j = 1, 2, …,

with initial condition z_{j}(0) = z_{j0}, j ≥ 1, where (z_{j0}) ∈ ℓ_{2}(ℂ) and z(t) = (z_{j}(t))_{j} ∈ ℓ_{2}(ℂ) for each t ∈ (t_{min}, t_{max}). Clearly, z_{j0} = ⟨u_{0}, φ_{j}⟩ = ∫_{Ω} u_{0}φ_{j}dx and ∥ · ∥ is a standard Euclid norm. For given u_{0} ∈ L^{2}(Ω, ℂ), we interpret NLS equation (1)_{1} as the lattice Schrödinger equation (3). To any v ∈ L^{2}(Ω, ℂ) one can assign z = (z_{j})_{j} ∈ ℓ_{2}(ℂ), where v = ∑_{j} z_{j}φ_{j} is the Fourier expansion of v, and this assignment is an isometric isomorphism.

Here, we note that the NLS Cauchy problem (1) is a natural extension of the Hopf bifurcation model for planar systems. The Hopf bifurcation is connected to 1-parameter families of vector fields where a limit cycle surrounding a singular point is born. It is well known that the Hopf bifurcation occurs at an equilibrium point x_{0} of a planar system ẋ = f(x, a_{0}) depending on a parameter a_{0} ∈ ℝ when the matrix Df(x_{0}, a_{0}) has a pair of pure imaginary eigenvalues, see [21, pg. 314.]. In this sense we say that the point x_{0} is the weak focus of the system. On the other side, if the eigenvalues are such that both their real and imaginary parts differ from zero, we are speaking about the strong focus.

Namely, when z = (z_{1}, 0, 0, …), and assuming that λ_{1} = 1, γ = 1, the system (3) reduces to the classical Hopf bifurcation system in the plane:

(4)  \{ \begin{matrix}ẋ & = & y − x(x^{2} + y^{2} + a_{0}) \\ ẏ & = & −x − y(x^{2} + y^{2} + a_{0}), \end{matrix}

where we denote x(t) = Re z(t), y(t) = Im z(t), real and imaginary part of the complex number z(t), respectively. System (4) written using the polar coordinates (r, θ) reads

ṙ = r(r^{2} + a_{0}), θ̇ = −1. (5)

and can be explicitly solved. Since θ̇ ≠ 0, the origin r = 0 is the only critical point. Since θ̇ < 0 the flow is always clockwise. The phase portrait of (4) for a_{0} < 0, a_{0} = 0 and a_{0} > 0 is shown by Figure 1. Viewing a_{0} as the bifurcation parameter, we point out the following cases in which the stability of the origin changes, i.e. the bifurcation occurs:

(i) For a_{0} > 0 the origin is unstable strong focus and for all trajectories we have (t_{min}, t_{max}) = (−∞, t_{0}) with t_{max} = t_{0} > 0.

(ii) For a_{0} = 0 the origin is unstable weak focus, and (t_{min}, t_{max}) = (−∞, t_{0}), t_{0} > 0. (iii) For a_{0} < 0 the origin is a stable strong focus. The limit cycle is the circle of

radius r = \sqrt{−a_{0}}, which is unstable. For trajectories inside the circle the solutions EJQTDE, 2010 No. 60, p. 4

Figure 1. Phase portrait of the classical Hopf bifurcation system (4).

are global, i.e. (t_{min}, t_{max}) = (−∞, ∞), while for those outside the circle we have (t_{min}, t_{max}) = (t_{0}, ∞) with t_{0} < 0.

Next, let us consider the Hopf bifurcation in a complex system (3), but in finitedimensional phase space. We start with two-dimensional complex system

(6)  ż_{j} = −iλ_{j}z_{j} − γz_{j}(|z_{1}|^{2} + |z_{2}|^{2} + a_{0}), j = 1, 2,

with the corresponding initial conditions z_{j}(0) = z_{j0}, where a_{0} is the bifurcation parameter. To simplify, for the moment we assume that γ ∈ ℝ and γ > 0. Near a_{0} = 0 we have the qualitative change of the behavior of the system. Indeed, using the polar coordinates the system (6) can be written as

ṙ_{j} = γr_{j}(r_{1}^{2} + r_{2}^{2} + a_{0}), θ̇_{j} = −λ_{j}, j = 1, 2,

EJQTDE, 2010 No. 60, p. 5 where z_{j} = r_{j} exp(iθ_{j}). For a_{0} < 0 an invariant set, 3-sphere

\sqrt{−a_{0}}S^{3} = \{(z_{1}, z_{2}) ∈ ℝ^{4} : |z_{1}|^{2} + |z_{2}|^{2} = −a_{0}, \} (7)

in ℝ^{4} is born, where S^{3} is the unit sphere in ℝ^{4}. In analogy with the case of ℝ^{2}, we call this phenomenon the Hopf bifurcation in ℝ^{4}. Clearly, if a_{0} → 0^{−}, then the sphere shrinks to the origin.

Moreover, if r_{1}^{2} + r_{2}^{2} < −a_{0}, then ṙ_{j} < 0, and r(t) is decreasing as t → ∞. The α-limit set of the trajectory is the origin, while the ω-limit set is a subset of the sphere (7). More precisely, the corresponding limit set is the subset of the 2-torus, contained in the 3-sphere (7). The torus has the form

r_{1}S^{1} × r_{2}S^{1} ⊂ \sqrt{−a_{0}}S^{3},

where r_{1,2} are positive scaling numbers depending on the initial point, r_{1}^{2}+r_{2}^{2} = −a_{0}, and S^{1} is the unit circle in the complex plane ℂ. Clearly, if r_{1}^{2} + r_{2}^{2} > −a_{0}, and r(t) monotonically increases, then the corresponding trajectories converge to the sphere and their ω-limit set is a subset of a 2-torus contained in the 3-sphere (7).

In the similar way the Hopf bifurcation for the following system of k complex equations:

(8)  ż_{j} = −iλ_{j}z_{j} − γz_{j}(|z_{1}|^{2} + ⋯ + |z_{k}|^{2} + a_{0}), j = 1, 2, …, k,

can be understood. An invariant (2k − 1)-sphere is born in ℂ^{k} when a_{0} < 0, defined by |z_{1}|^{2}+⋯+|z_{k}|^{2} = −a_{0}. For the general theory of bifurcation problems in finite-dimensional dynamical systems see Guckenheimer and Holmes [13].

1.3. Hopf bifurcation in an infinite-dimensional phase space. Since we are interested in the phenomenon of the Hopf bifurcation in an infinite-dimensional complex system (3), we start with the description of the Hopf bifurcation for an ODE defined in a Banach space X (real or complex):

(9)  u̇ = F (u, a_{0}), u(0) = u_{0}.

Here F : D × ℝ → X is a given map, where D is a dense subspace of X, and the initial value u_{0} ∈ D is prescribed. For our purposes the mapping F is usually of the form

F (u, a_{0}) = Au + f(u, a_{0}),

where A : D ⊆ X → X is a second order differential operator, f: X × ℝ → X continuous such that f(u, 0) = o(u) as u → 0, and a_{0} is the bifurcation parameter. Additionally we assume that for all a_{0} we have F (0, a_{0}) = 0 so that u = 0 is an equilibrium point. For each initial point u_{0} in a neighbourhood of 0 ∈ D let a trajectory

Γ(u_{0}) = {u(t) ∈ X: t ∈ (t_{min}, t_{max}), u(0) = u_{0}} (10)

of (9) be defined on an unbounded interval (t_{min}, t_{max}) containing the origin.

Definition 1. We say that a_{0} = 0 is the point of Hopf bifurcation for the system (9) if the following conditions are fulfilled:

EJQTDE, 2010 No. 60, p. 6 (i) For a_{0} ≥ 0 small enough the system is unstable near the origin, and for u_{0} from a neighbourhood of the origin we have (t_{min}, t_{max}) = (−∞, t_{0}), t_{max} = t_{0} > 0.

(ii) For a_{0} < 0 with |a_{0}| small enough, the origin is stable, and an unstable invariant set S(a_{0}) ⊂ X for (9) is born near the origin. There exists an open neighbourhood U(a_{0}) of the origin such that its boundary is S(a_{0}), and for any u_{0} ∈ \overline{U(a_{0})} the corresponding solution is global, i.e. (t_{min}, t_{max}) = (−∞, ∞), while for u_{0} ∈ B \ \overline{U(a_{0})} we have that (t_{min}, t_{max}) = (−∞, t_{0}) with t_{0} > 0.

Definition 1 is analogous if the signs of a_{0} are reversed, in which case the stability should be reversed as well as semi-infinite time intervals. Somewhat loosely we say that the Hopf bifurcation occurs if an invariant set is born near the origin when a_{0} passes through the value of 0, and the origin changes from stable to unstable or vice versa.

Returning to the NLS Cauchy problem (1), the Hopf bifurcation consists in the birth of an invariant sphere in the space L^{2}(Ω) when a_{0} < 0. Here we introduce some notation. For a_{0} < 0 we define R = \sqrt{−a_{0}} and let

B_{R}(0) = {v ∈ L^{2}(Ω): ∥v∥_{L^{2}(Ω)} < R}, S_{R}(0) = ∂B_{R}(0), (11)

be the ball and the sphere in L^{2}(Ω), respectively. In the same way, when a_{0} < 0 for the corresponding system (3) in ℓ_{2}(ℂ), the Hopf bifurcation consists in the birth of an invariant sphere in the space ℓ_{2}(ℂ). Since (8) is also a special case of the NLS Cauchy problem (1), corresponding to the case when u_{0} ∈ span {φ_{1}, …, φ_{k}}, we see that we can expect trajectories of (1) in the Hilbert space L^{2}(Ω, ℂ), which oscillate at infinitely many scales. We can achieve this by choosing u_{0} so that ⟨u_{0}, φ_{j}⟩ ≠ 0 for infinitely many j’s.

Furthermore, we are interested in Hopf bifurcations from the point of view of fractal geometry. For that purpose, in the next subsection firstly we review some standard notation and definitions from fractal geometry and Sobolev spaces.

1.4. Notation and definitions. Let A be a bounded set in ℝ^{k}, and let d(x, A) be Euclidean distance from x to A. Then the Minkowski sausage A_{ε} is A_{ε} := {y ∈ ℝ^{k}d(y, A) < ε}, a term coined by B. Mandelbrot. By lower s-dimensional Minkowski content of A, s ≥ 0, we mean the following:

(12)  M_{∗}^{s}(A) := lim inf_{ε→0} \frac{|A_{ε}|}{ε^{k−s}} ,

where | · | is N-dimensional Lebesgue measure. Analogously for the upper s-dimensional Minkowski content of A. The lower box dimension of A is defined by

\underline{dim}_{B}A = inf\{s > 0 : M_{∗}^{s}(A) = 0\},

and analogously the upper box dimension \overline{dim}_{B}A. For various properties of fractal dimensions see Falconer [11].

EJQTDE, 2010 No. 60, p. 7

If A is such that \underline{dim}_{B}A = \overline{dim}_{B}A, then the common value is denoted by d := dim_{B} A, and is called the box dimension of A. Furthermore, if both the upper and lower ddimensional Minkowski contents of A are different from 0 and ∞, we say that the set A is Minkowski nondegenerate. If in addition to this we have M_{∗}^{d}(A) = M^{∗d}(A) =: M^{d}(A) ∈ (0, ∞), then A is said to be Minkowski measurable. The notion of Minkowski content appears for example in the study of fractal drums and fractal strings, see He and Lapidus [14], Lapidus and Frankenhuysen [18]. Furthermore, Minkowski content is essential for understanding some singular integrals, see [25].

If A is a subset of an infinite-dimensional vector space X, we say that dim_{B} A = ∞ if there exists an increasing sequence of finite-dimensional subspaces X_{k} of X such that dim_{B}(A ∩ X_{k}) → ∞ as k → ∞. We need this definition in Theorem 6(b).

We deal with the Sobolev spaces H_{0}^{1}(Ω, ℂ) and H_{0}^{1}(Ω, ℂ) ∩ H^{2}(Ω, ℂ) (in the sequel we omit ℂ) equipped with the corresponding norms defined by

(13)  ∥u_{0}∥_{H_{0}^{1}}^{2} = ∑_{j} λ_{j}|⟨u_{0}, φ_{j}⟩|^{2}, ∥u_{0}∥_{H_{0}^{1}∩H^{2}}^{2} = ∑_{j} λ_{j}^{2}|⟨u_{0}, φ_{j}⟩|^{2}.

See Henry [1, 15].

All the results of this paper hold if in (1) we have −Δ instead of Δ. The j-th component of trajectory, viewed as a spiral in ℂ, only changes the orientation for each j from negative to positive, see (18) below.

2. Well-posedness and stability of solutions

2.1. Explicit solutions of the NLS problem. We consider the NLS initial-boundary value problem (1). For some special values of parameters γ and a_{0} it is possible to find explicit solutions of problem (1) and then to consider their qualitative properties. In order to calculate directly the explicit solutions of problem (1), we need the following lemma.

Lemma 2. Let u be a solution of the NLS Cauchy problem (1) in the form (2). Furthermore, let ρ(t) = ∥u(t)∥_{L^{2}}, z_{j}(t) = r_{j}(t) exp(iθ_{j}(t)), and V (ρ) = ρ^{2} + a_{0}. Then

ρ̇ = −γ_{1}ρV (ρ)
(14)  ṙ_{j} = −γ_{1}r_{j}V (ρ), j ∈ ℕ
θ̇_{j} = −λ_{j} − γ_{2}V (ρ), j ∈ ℕ,

where γ = γ_{1} + iγ_{2}. Furthermore, for u_{0} ≠ 0 we have

(15)  r_{j}(t) = \frac{|⟨u_{0}, φ_{j}⟩|}{∥u_{0}∥_{L^{2}}} ρ(t).

Proof. In Section 1, we showed that the NLS Cauchy problem (1) can be written in the form on an infinite-dimensional ODE system

ż_{j} = −iλ_{j}z_{j} − γV (ρ)z_{j}, j ∈ ℕ, (16)

EJQTDE, 2010 No. 60, p. 8 where z_{j} = r_{j} exp(iθ_{j}). The expressions (14)_{2} and (14)_{3} follow easily by multiplying the expression (16) by exp(−iθ_{j}). The first equation in (14) we get directly from the formal calculation. Namely,

2ρρ̇ = \frac{d}{dt} ∫_{Ω} |u(t)|^{2}dx = ∫_{Ω} \frac{d}{dt}(u(t)\overline{u}(t)) dx = ∫_{Ω}(u̇(t)\overline{u}(t) + u(t) ˙\overline{u}(t)) dx
= i ∫_{Ω}(Δu(t) \overline{u}(t) − u(t) Δ\overline{u}(t)) dx − 2γ_{1} ∫_{Ω} |u(t)|^{2}dx · V (∥u(t)∥_{L^{2}})
= −2γ_{1}ρ^{2}V (ρ).

By dividing the first two expressions given in (14) we obtain

\frac{ṙ_{j}}{r_{j}} = \frac{ρ̇}{ρ},

hence r_{j} = C_{j}ρ, where C_{j} is a constant depending on the initial value u_{0}. Using the expression (2), we obtain that

c_{j} = \frac{|⟨u_{0}, φ_{j}⟩|}{∥u_{0}∥_{L^{2}}}e^{−iθ_{j}(0)},

where ⟨u_{0}, φ_{j}⟩ = ∫_{Ω} u_{0}φ_{j}dx ∈ ℂ. □

Next, we use the results given by lemma 2 in order to find the explicit solutions of the NLS Cauchy problem (1) for some special values of the parameters γ and a_{0}. We use the notation ρ_{0} = ∥u_{0}∥_{L^{2}}. For the sake of simplicity we take γ_{2} = 0, i.e. γ ∈ ℝ.

Case 1: a_{0} ∈ ℝ, γ_{1} = 0.

¿From lemma 2 directly follows that for γ_{1} = 0 it holds ∥u(t)∥_{L^{2}} = C(u_{0}), and moreover, r_{j}(t) = C_{j}(u_{0}) for each j ∈ ℕ.

Case 2: a_{0} = 0, γ_{1} ≠ 0.

In this case V (ρ) = ρ^{2} and the explicit solution of equation (14)_{1} is given by

(17)  ρ(t) = (2γ_{1}t + ρ_{0}^{−2})^{−1/2}.

¿From (15) and (14)_{3} using (17), and noting θ_{j}(0) = arg⟨u_{0}, φ_{j}⟩ we obtain

(18)  \{ \begin{matrix}r_{j}(t) & = & \frac{|⟨u_{0},φ_{j}⟩|}{ρ_{0}}(2γ_{1}t + ρ_{0}^{−2})^{−1/2}, \\ θ_{j}(t) & = & −λ_{j}t − \frac{γ_{2}}{2γ_{1}} ln |2γ_{1}t + ρ_{0}^{−2}| + arg⟨u_{0}, φ_{j}⟩, \end{matrix}

where z_{j}(t) = r_{j}(t) exp(iθ_{j}(t)).

In this way, using the decomposition (2) we have the explicit solution of NLS problem (1)

(19)  u(t, x) = (2γ_{1}t + ∥u_{0}∥_{L^{2}}^{−2})^{−1/2} ∑_{j=1}^{∞} \frac{⟨u_{0}, φ_{j}⟩}{∥u_{0}∥_{L^{2}}} e^{−iλ_{j}t} φ_{j}(x),

where u_{0} is a given initial function. Here we notice that the sign of the parameter γ_{1} affects the maximal interval of the solution. More precisely, following the terminology introduced in Cazenave [2, Remark 3.1.6(ii)], we say that the solutions given by formula (19) are positively (negatively) global if γ_{1} > 0 (γ_{1} < 0). The solutions are global for γ_{1} = 0 for any initial value u_{0}, while for γ_{1} ≠ 0 the global solution exists only when u_{0} = 0.

Case 3: a_{0} ≠ 0, γ_{1} ≠ 0.

In this case one has V (ρ) = ρ^{2} + a_{0} and we obtain the Bernoulli equation ρ̇ = −γ_{1}ρV (ρ) with the solution

(20)  ρ(t) = [(ρ_{0}^{−2} + a_{0}^{−1})e^{(2γ_{1}a_{0}t)} − a_{0}^{−1})]^{−1/2}.

We obtain the analogous series representation of solution u(t, x) of (1) as in (19), assuming again that γ_{2} = 0:

(21)  u(t, x) = ((ρ_{0}^{−2} + a_{0}^{−1})e^{2γ_{1}a_{0}t} − a_{0}^{−1})^{−1/2} ∑_{j=1}^{∞} \frac{⟨u_{0}, φ_{j}⟩}{∥u_{0}∥_{L^{2}}} e^{−iλ_{j}t} φ_{j}(x).

The ability to calculate the explicit solution of NLS boundary-value problem (1), given by formulas (19) and (21), guarantees its uniqueness. More precisely, it is obvious that the following result is true.

Proposition 3. Let u_{0} ∈ L^{2}(Ω) be the initial function for NLS initial-boundary value problem (1), where a_{0} ∈ ℝ is the bifurcation parameter. For any u_{0} ∈ L^{2}(Ω), NLS initialboundary value problem (1) possesses a unique solution of the form (2), with z_{j}(t) ∈ ℓ_{2}(ℂ) and z_{j}(·) of class C^{1}. Moreover, the solution is represented by formula (19) and (21) for a_{0} = 0 and a_{0} ≠ 0, respectively.

The explicit formulas (19) and (21) for the solution of the NLS problem (1) enable us to express the corresponding norms of the solution in the case when u_{0} ∈ H_{0}^{1}(Ω) or u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω), respectively. Namely, it follows

∥u(t)∥_{H_{0}^{1}}^{2} = ρ(t)^{2} ∑_{j} λ_{j} \frac{|⟨u_{0}, φ_{j}⟩|^{2}}{∥u_{0}∥_{L^{2}}^{2}} = ρ(t)^{2} \frac{∥u_{0}∥_{H_{0}^{1}}^{2}}{∥u_{0}∥_{L^{2}}^{2}} .,

where ρ(t) is defined by (17) or (20) if a_{0} = 0 or a_{0} ≠ 0 respectively. Similarly for u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω) we have

∥u(t)∥_{H_{0}^{1}∩H^{2}} = ρ(t)\frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{∥u_{0}∥_{L^{2}}} .

Next, let us consider formula (17) again. It is obvious that for t → t_{min}^{+} where t_{min} = −(2γ_{1}ρ_{0}^{2})^{−1} one has ∥u(t)∥_{L^{2}(Ω)} → ∞, i.e. we have the blow up of the solution. Similar conclusion holds for a_{0} ≠ 0 and the expression (20).

EJQTDE, 2010 No. 60, p. 10

Finally, we note that directly from formulas (19) and (21) follows the invariance property of the solutions of the NLS problem. More precisely, if U_{0} is the span of a given subset of \{φ_{j} : j ≥ 1\} ⊂ L^{2}(Ω), then the assumption u_{0} ∈ \overline{U}_{0} implies that u(t) ∈ \overline{U}_{0} for all t > 0, where the closure is taken in L^{2}(Ω). Moreover, the invariance property can be reformulated as follows: each subspace \overline{U}_{0}, where U_{0} is spanned by a subset of φ_{j}-s, is invariant for the nonlinear evolution operator T (t), T (t)u_{0} = u(t), see (19), associated with the problem (1): T (t)\overline{U}_{0} ⊆ \overline{U}_{0}. In other words, a trajectory that starts in \overline{U}_{0} remains in this space forever. In particular, if u_{0} ∈ span {φ_{1}, …, φ_{k}}, then Γ(u_{0}) ⊂ span {φ_{1}, …, φ_{k}}. This means that if ⟨u_{0}, φ_{j}⟩ = 0 for all but finitely many j’s, then (1) is essentially a finite-dimensional problem, which can be viewed as (8).

2.2. Well posedness and stability. The following proposition gives some stability results of the solution of NLS boundary-value problem (1) with respect to the value of the bifurcation parameter a_{0}. Again, for the sake of simplicity we assume that γ_{2} = 0 and γ_{1} > 0. For γ_{1} < 0 time intervals of the form (t_{min}, ∞) should be changed to (−∞, t_{max}). The notation used in the following proposition (the ball and the sphere in L^{2}(Ω)) we introduced in (11). Directly from the expressions (17) and (20) follow the power and the exponential rate of the convergence of ∥u(t)∥_{L^{2}} to the origin in L^{2}(Ω) (as the fixed point for problem (1)), respectively. More precisely, the following proposition holds. Proposition 4. (Hopf bifurcation for Schrödinger problem) Assume that γ = γ_{1} > 1 and let u_{0} ∈ L^{2}(Ω) be the initial function for NLS initial-boundary value problem (1), where a_{0} ∈ ℝ is the bifurcation parameter.

(i) For a_{0} > 0 then the origin is exponentially stable with respect to L^{2}-topology for any u_{0} ≠ 0.

(ii) For a_{0} = 0 the origin is power stable in L^{2}(Ω), with power α = 1/2.

(iii) For a_{0} < 0 let us denote R = \sqrt{−a_{0}}. Then we distinguish the following cases:

(a) If u_{0} ∈ B_{R}(0) then the origin is exponentially unstable and the solutions are global with ∥u(t)∥_{L^{2}} → R as t → ∞.

(b) If u_{0} ∈ L^{2}(Ω)\B_{R}(0) then the solution u(t) is positively global and ∥u(t)∥_{L^{2}} → R as t → ∞.

(c) If u_{0} ∈ S_{R}(0) then also u(t) ∈ S_{R}(0), so the sphere S_{R}(0) is an invariant attractor.

The continuous dependence on the initial condition, regularity and the continuous dependence on bifurcation parameter for NLS boundary-value problem (1) when u_{0} ∈ L^{2}(Ω) is given by the following proposition. Here, we point out that the same qualitative properties valid if u_{0} ∈ H_{0}^{1}(Ω) or H_{0}^{1}(Ω) ∩ H^{2}(Ω). The interval of the existence of u(t) we note by I = (t_{min}, t_{max}). Proposition 5. (a) (continuous dependence on initial condition) If v_{0} → u_{0} in L^{2}(Ω) then for each fixed t ∈ I we have that v(t) → u(t) in L^{2}(Ω). Moreover, the convergence is uniform on each compact interval in I.

EJQTDE, 2010 No. 60, p. 11 (b) (regularity) If u_{0} ∈ L^{2}(Ω) then u ∈ C(I, L^{2}(Ω)).

(c) (continuous dependence on bifurcation parameter) Let u_{0} ∈ L^{2}(Ω) be fixed, and let u(t) and u_{a_{0}}(t) be defined by (19) and (21) respectively. Then for any t ∈ I,

∥u_{a_{0}}(t) − u(t)∥_{L^{2}} → 0 as a_{0} → 0,

where I is the interval of existence of u(t), Moreover, the convergence is uniform on compact intervals J contained in I.

Proof. (a) If v_{0} = 0 then the claim follows easily from proposition 4. Let v_{0} ≠ 0 and u_{0} → v_{0}. Denoting û_{0} = u_{0}/∥u_{0}∥_{L^{2}} and using |a+b|^{2} ≤ 2|a|^{2} +2|b|^{2} for a, b ∈ ℂ, we obtain:

∥u(t) − v(t)∥_{L^{2}}^{2} = ∑_{j} |⟨û_{0}, φ_{j}⟩ρ_{u_{0}}(t) − ⟨v̂_{0}, φ_{j}⟩ρ_{v_{0}}(t)|^{2}
= ∑_{j} |(⟨û_{0}, φ_{j}⟩ − ⟨v̂_{0}, φ_{j}⟩)ρ_{u_{0}}(t) + ⟨v̂_{0}, φ_{j}⟩(ρ_{u_{0}}(t) − ρ_{v_{0}}(t))|^{2}
≤ 2ρ_{u_{0}}(t)^{2} ∑_{j} |⟨û_{0} − v̂_{0}, φ_{j}⟩|^{2} + 2|ρ_{u_{0}}(t) − ρ_{v_{0}}(t)|^{2} ∑_{j} |⟨v̂_{0}, φ_{j}⟩|^{2}
= 2ρ_{u_{0}}(t)^{2}∥û_{0} − v̂_{0}∥_{L^{2}}^{2} + 2|ρ_{u_{0}}(t) − ρ_{v_{0}}(t)|^{2}∥v̂_{0}∥_{L^{2}}^{2}.

Therefore, since u_{0} → v_{0} in L^{2}(Ω) it follows that for each fixed t we have ∥u(t)−v(t)∥_{L^{2}} → 0.

(b) Assuming that u_{0} ∈ L^{2}(Ω), we first write u(t) = ρ(t)S(t), where we note S(t) = ∑_{j=1}^{∞} \frac{⟨u_{0},φ_{j}⟩}{∥u_{0}∥_{L^{2}}}e^{−iλ_{j}t}φ_{j}(x). We have u(t) − u(s) = (ρ(t) − ρ(s))S(t) + ρ(s)(S(t) − S(s)), and from this

(22)  ∥u(t) − u(s)∥_{L^{2}}^{2} ≤ 2(ρ(t) − ρ(s))^{2} + 2ρ(s)^{2} ∑_{j=1}^{∞} |⟨û_{0}, φ_{j}⟩|^{2}|e^{−iλ_{j}t} − e^{−iλ_{j}s}|^{2}.

Since ρ(t) is uniformly continuous, one has that ρ(t) → ρ(s) for t ≥ s and it suffices to show that, for given ε the expression

∑_{j=1}^{∞} |⟨û_{0}, φ_{j}⟩|^{2}|e^{−iλ_{j}t} − e^{−iλ_{j}s}|^{2}

can be made less than ε if |t − s| is small enough. For any m > 1, the sum is less than or equal to:

∑_{j=1}^{m} |e^{−iλ_{j}t} − e^{−iλ_{j}s}|^{2} + 4 ∑_{j=m}^{∞} |⟨û_{0}, φ_{j}⟩|^{2},

since |⟨û_{0}, φ_{j}⟩| ≤ 1 and |e^{−iλ_{j}t} − e^{−iλ_{j}s}| ≤ 2. We choose m = m(u_{0}, ε) large enough so that the second sum is ≤ ε/8. It is clear that the first sum can be made ≤ ε/2 when |t − s| ≤ δ for δ = δ(ε, m) > 0 small enough.

(c) The claim follows from ∥u_{a_{0}}(t) − u(t)∥_{L^{2}} = |ρ_{a_{0}}(t) − ρ(t)| → 0 uniformly in t ∈ J as a_{0} → 0, since a^{−1}[exp(2γ_{1}a_{0}t) − 1] → 2γ_{1}t uniformly in t ∈ J as a_{0} → 0. □

EJQTDE, 2010 No. 60, p. 12

3. Compactness and non-rectifiability of trajectories

3.1. Compactness. Since we consider trajectories in infinite-dimensional spaces, it is not at all clear if they are compact sets, or even rectifiable. Using the notation given in section 1.2, the solution of NLS problem (1) can be written in the following way

u(t) = ρ(t) ∑_{j}⟨û_{0}, φ_{j}⟩e^{−iλ_{j}t}φ_{j}(x), (23)

where u_{0} is a given initial function. By Γ(u_{0}) = {u(t): t ≥ t_{0}} we note the trajectories of solutions. The notation (23) enables us to reduce the problem of compactness of the trajectories to the problem of the compactness of the bounded sets in ℓ_{2}(ℂ). In this sense, we review the following characterization of relatively compact sets in ℓ_{2}(ℂ) (Weidmann [24, p. 135]):

A subset Y of ℓ_{2}(ℂ) is relatively compact in ℓ_{2}(ℂ) if and only if Y is bounded and for every ε > 0 there exists j_{0} ∈ ℕ such that for all sequences (f_{j}) in Y the following condition is fulfilled:

(24)  ∑_{j≥j_{0}} |f_{j}|^{2} ≤ ε.

The following theorem establishes not only compactness of individual trajectories Γ(u_{0}) of solutions of the NLS Cauchy problem (1) in L^{2}(Ω), but also for some bundles of trajectories. If a trajectory generated by u_{0} is positively global, defined for t ∈ (t_{min}, ∞), then by Γ(u_{0}) we denote its part corresponding to t ≥ 0. We do analogously for negatively global trajectories.

For a given nonempty base set of initial functions A ⊂ L^{2}(Ω) we can define the corresponding bundle of trajectories by

Γ(A) = ⋃_{v_{0}∈A} Γ(v_{0}).

The following theorem provides some sufficient conditions on A that ensure compactness of Γ(A). To simplify, we assume that a_{0} = 0 and γ_{1} > 0 in (1).

Theorem 6. (a) For any u_{0} ∈ L^{2}(Ω) the corresponding trajectory Γ(u_{0}) of (1) starting with t_{0} = 0 is relatively compact in L^{2}(Ω).

(b) Let u_{0} ∈ L^{2}(Ω) be given and define

A(u_{0}) = {v_{0} ∈ L^{2}(Ω) : |⟨v_{0}, φ_{j}⟩| ≤ |⟨u_{0}, φ_{j}⟩|, ∀j}.

Then we have A(u_{0}) = Γ(A(u_{0})), and this set is compact in L^{2}(Ω).

(c) If u_{0}, v_{0} ∈ L^{2}(Ω) are given, then the bundle Γ([u_{0}, v_{0}]) generated by the line segment [u_{0}, v_{0}] = {(1 − λ)u_{0} + λv_{0} : 0 ≤ λ ≤ 1} is relatively compact in L^{2}(Ω). More generally, if A is a finite set in L^{2}(Ω) and conv A its convex hull, then Γ(conv A) is relatively compact.

EJQTDE, 2010 No. 60, p. 13 Proof. (a) Let u_{0} ≠ 0 be a fixed initial function (for u_{0} = 0 the claim is trivial, since Γ(0) = {0}). The trajectory Γ(u_{0}) is identified with the corresponding trajectory in ℓ_{2}(ℂ), that is

u(t) = ∑_{j} z_{j}(t)φ_{j} ↦ (z_{j}(t))_{j} ∈ ℓ_{2}(ℂ).

where u(t) is generated by u_{0} and

z_{j}(t) = ρ(t)⟨û_{0}, φ_{j}⟩e^{−iλ_{j}t}.

Now, we show that the set {(z_{j}(t))_{j} : t ≥ 0)} is relatively compact in ℓ_{2}(ℂ). Using (18) and (17), for all t ≥ 0 we have that

∑_{j≥j_{0}} |z_{j}(t)|^{2} = ρ^{2}(t) ∑_{j≥j_{0}} r_{j}(t)^{2} = \frac{1}{1 + 2γ_{1}ρ_{0}^{2}t} ∑_{j≥j_{0}} |⟨u_{0}, φ_{j}⟩|^{2}
≤ ∑_{j≥j_{0}} |⟨u_{0}, φ_{j}⟩|^{2} ≤ ε,

that is, condition (24) is fulfilled provided j_{0} is large enough. The case of a_{0} ≠ 0 is treated similarly.

(b) Let us first prove that A(u_{0}) = Γ(A(u_{0})). The inclusion ⊆ is clear. To prove the converse inclusion, let v(t) ∈ Γ(A(u_{0})), where v(t) is a trajectory generated by v_{0} ∈ A(u_{0}). Then

|⟨v(t), φ_{j}⟩| = |ρ(t)⟨v̂_{0}, φ_{j}⟩| = \frac{ρ(t)}{∥v_{0}∥_{L^{2}}} |⟨v_{0}, φ_{j}⟩|
≤ |⟨v_{0}, φ_{j}⟩| ≤ |⟨u_{0}, φ_{j}⟩|,

where we used the monotonicity of ρ(t) for γ_{1} > 0, so that for t ≥ 0 we have ρ(t) ≤ ρ(0) = ∥u_{0}∥_{L^{2}}. Hence v(t) ∈ A(u_{0}).

The compactness of A(u_{0}) is proved using a slight change in the proof of (a). Denoting by A′(u_{0}) the set in ℓ_{2}(ℂ) corresponding to A(u_{0}) in L^{2}(Ω), then

(25)  A′(u_{0}) = ∏_{j} \overline{B_{r_{j}}(0)},

where r_{j} = |⟨u_{0}, φ_{j}⟩| and B_{r_{j}}(0) is the open disk of radius r_{j} in ℂ = ℝ^{2} imbedded into the j-th component of ℓ_{2}(ℂ). To prove (25), it suffices to note that if v = ∑_{j} z_{j}φ_{j}, then z = (z_{j})_{j} ∈ A(u_{0}) if and only if z_{j} ∈ \overline{B_{r_{j}}(0)} for all j.

(c) Let r_{j} = |⟨u_{0}, φ_{j}⟩| and s_{j} = |⟨v_{0}, φ_{j}⟩|. Defining w = ∑_{j} max{r_{j}, s_{j}}φ_{j} it is clear that w ∈ L^{2}(Ω) since

∑_{j} max{r_{j}, s_{j}}^{2} ≤ ∑_{j}(r_{j}^{2} + s_{j}^{2}) = ∥u_{0}∥_{L^{2}}^{2} + ∥v_{0}∥_{L^{2}}^{2} < ∞.

Let us show that [u_{0}, v_{0}] ⊂ A(w). Indeed, taking v ∈ [u_{0}, v_{0}], for any j we have

|⟨v, φ_{j}⟩| = |⟨(1 − λ)u_{0} + λv_{0}, φ_{j}⟩| ≤ (1 − λ)r_{j} + λs_{j} ≤ max{r_{j}, s_{j}} = |⟨w, φ_{j}⟩|.

Since Γ([u_{0}, v_{0}]) ⊂ Γ(A(w)) and due to the result given by (b), one concludes that the bundle Γ([u_{0}, v_{0}]) is compact in L^{2}(Ω). □

Remark 7. Similar results like those presented in Theorem 6 valid if u_{0} ∈ H_{0}^{1}(Ω) or u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω). More precisely, in then the trajectory Γ(u_{0}) is relatively compact in H_{0}^{1}(Ω) and H_{0}^{1}(Ω) ∩ H^{2}(Ω), respectively.

Indeed, the subspace H_{0}^{1}(Ω) of L^{2}(Ω) is isometrically isomorphic to the subspace ℓ′_{2} of ℓ_{2}(ℂ), consisting of all sequences z = (z_{j}) such that ∥z∥_{ℓ′_{2}}^{2} = ∑_{j} λ_{j}|z_{j}|^{2} < ∞. It is easy to see that the analogous characterization of compact sets as for ℓ_{2} in (a) holds also in ℓ′_{2} with respect to the new norm. Similarly for the subspace ℓ′′_{2} corresponding to H_{0}^{1}(Ω) ∩ H^{2}(Ω), with the norm ∥z∥_{ℓ′′_{2}}^{2} = ∑_{j} λ_{j}^{2}|z_{j}|^{2}.

3.2. Rectifiability. Assuming that t_{max}(u_{0}) = ∞ and ∥u(t)∥_{L^{2}} → 0 as t → ∞, we define the length of trajectory Γ(u_{0}) over the interval [0, t) by

l(u_{0}, t) = ∫_{0}^{t} ∥ż(s)∥_{ℓ_{2}} ds,

where z(t) = (z_{j}(t))_{j} ∈ ℓ_{2}(ℂ) is isometrically assigned to u(t) ∈ L^{2}(Ω), generated by u_{0} ∈ L^{2}(Ω), see (1), (2) and (3). We consider rectifiability and nonrecitifiability of Γ(u_{0}) only for t ∈ [0, ∞) or t ∈ (−∞, 0].

Moreover, we are interested in the asymptotic behaviour of l(u_{0}, t) as t → ∞. For that purpose, we introduce here some notation. We say that:

• f(t) ∼ g(t) as t → ∞ if lim_{t→∞} \frac{f(t)}{g(t)} = 1.

• f(t) ≃ g(t) as t → ∞ if lim_{t→∞} \frac{f(t)}{g(t)} < ∞ .

The following theorem states the dependence of the (non)-rectifiability of trajectories Γ(u_{0}) near the origin of the NLS initial-boundary value problem (1) due to the value of the bifurcation parameter a_{0}.

Theorem 8. Let Γ(u_{0}) be the trajectory of the solution of the NLS initial-boundary value problem (1) and let be γ_{1} > 0 and u_{0} ≠ 0. Each trajectory Γ(u_{0}) of (1) near the origin is nonrectifiable for a_{0} = 0 and rectifiable for a_{0} > 0. Moreover, for a_{0} = 0 and u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω) we have the following asymptotic result:

(26)  l(u_{0}, t) ∼ \frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{∥u_{0}∥_{L^{2}}} (2γ_{1}^{−1}t)^{1/2} as t → ∞.

In particular, l(u_{0}, t) ≃ t^{1/2} as t → ∞.

EJQTDE, 2010 No. 60, p. 15

If a_{0} < 0, then for u_{0} ∈ B(R) the trajectory is rectifiable for t ∈ (−∞, 0], and nonrectifiable for t ∈ [0, ∞). If u_{0} ∉ B(R), the trajectories are nonrectifiable. In both of these nonrectifiable cases we have l(u_{0}, t) ≃ t as t → ∞.

Proof. Assume that a_{0} = 0 and u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω). For the sake of simplicity we assume that γ_{2} = 0, so that θ̇_{j} = −λ_{j}, see (18) (the case of γ_{2} ≠ 0 can be treated with slight modifications of the proof). Using z_{j} = r_{j} exp(iθ_{j}), |ż_{j}|^{2} = ṙ_{j}^{2} + r_{j}^{2}θ̇_{j}^{2}, ant expressions (18) and (13), we obtain the following estimate for l(u_{0}, t):

l(u_{0}, t) = ∫_{0}^{t} (∑_{j=1}^{∞} |ż_{j}|^{2})^{1/2} ds = ∫_{0}^{t} (∑_{j} ṙ_{j}^{2} + ∑_{j} r_{j}^{2}θ̇_{j}^{2})^{1/2} ds

(27)  = ∫_{0}^{t} (γ_{1}^{2}(2γ_{1}s + ρ_{0}^{−2})^{−3} + (2γ_{1}s + ρ_{0}^{−2})^{−1} \frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}^{2}}{ρ_{0}^{2}} )^{1/2} ds

≥ \frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{ρ_{0}} ∫_{0}^{t}(2γ_{1}s + ρ_{0}^{−2})^{−1/2}ds,

where ρ_{0} = ∥u_{0}∥_{L^{2}(Ω)}.

On the other hand, using inequality (a^{2} + b^{2})^{1/2} ≤ |a| + |b|, see (27), we obtain the estimate from below for l(u_{0}, t) which reads

l(u_{0}, t) ≤ γ_{1} ∫_{0}^{t}(2γ_{1}s + ρ_{0}^{−2})^{−3/2}ds + \frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{ρ_{0}} ∫_{0}^{t}(2γ_{1}s + ρ_{0}^{−2})^{−1/2}ds
≤ ρ_{0} + \frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{ρ_{0}} ∫_{0}^{t}(2γ_{1}s + ρ_{0}^{−2})^{−1/2}ds,

The claim in (26) follows by direct computation. The case when a_{0} ≠ 0 is treated similarly, using (21). □

Remark 9. If u_{0} ∈ L^{2}(Ω) \ (H_{0}^{1}(Ω) ∩ H^{2}(Ω)) then l(u_{0}, t) = ∞ for each t > 0. Indeed, in this case we have that ∥u_{0}∥_{H_{0}^{1}∩H^{2}} = ∑_{j} λ_{j}^{2}|⟨u_{0}, φ_{j}⟩|^{2} = ∞, and the claim follows from (27).

Remark 10. Let A be a bounded subset of H_{0}^{1}(Ω) ∩ H^{2}(Ω) such that 0 ∉ \overline{A}, with closure taken in this space. Let us consider the bundle of trajectories Γ(A), and let us define lower and upper length of bundle Γ(A) in time interval [0, t) by:

\underline{l}(A, t) = inf\{l(u_{0}, t) : u_{0} ∈ A\}, \overline{l}(A, t) = sup\{l(u_{0}, t) : u_{0} ∈ A\}.

Assuming that a_{0} = 0, γ_{1} > 0, u_{0} ≠ 0, it can be shown by reconsidering the proof of Theorem 8 that

\underline{l}(A, t) ≃ t^{1/2}, \overline{l}(A, t) ≃ t^{1/2} as t → ∞.

EJQTDE, 2010 No. 60, p. 16

4. Minkowski sequence associated to a trajectory

In this section we consider NLS boundary-initial value problem (1) for a_{0} = 0. Let Γ(u_{0}) be a trajectory generated by u_{0} ≠ 0, corresponding to t ≥ 0 if the solution u(t) is positively global, and to t ≤ 0 if it is negatively global. Denote by Γ_{j}(u_{0}) the orthogonal projection of Γ(u_{0}) ⊂ L^{2}(Ω) onto the φ_{j}-component, where φ_{j} are defined in section 1.2. Here, we recall that the solution of (1) can be written in form u(t, x) = ∑_{j=1}^{∞} z_{j}(t)φ_{j}(x). Now, Γ_{j}(u_{0}) can be viewed as a curve in the complex plane defined by z_{j}(t) = ⟨u(t), φ_{j}⟩ = r_{j}(t) exp(iθ_{j}(t)).

Figure 2 shows an example of Γ_{j}(u_{0}), j = 1, 2, 3, 4, where z_{j}(t) = (t + 1)^{−1/2}c_{j}e^{−iλ_{j}t} with c_{j} = ⟨u_{0}, φ_{j}⟩. For some chosen eigenvalues λ_{j} and Fourier coefficients c_{j} we plotted z_{j}(t) on the time interval [0, ∞). The trajectory t ↦ z(t) of the considered system in ℓ_{2}(ℂ) is equal to a sequence t ↦ (z_{j}(t))_{j∈ℕ}. In this way Figure 2 should indicate the projection of the considered trajectory into ℂ^{4}.

Figure 2. Projections Γ_{j}(u_{0}) of Γ(u_{0}), j = 1, 2, 3, 4.

The following result describes some properties of the sequence of d-dimensional Minkowski contents of curves Γ_{j}(u_{0}) for d = 4/3, j ∈ ℕ. Recall that if a set A ⊂ ℝ^{k} is such that d = dim_{B} A > 0, then M^{s}(A) = ∞ for 0 ≤ s < d and M^{s}(A) = 0 for s > d.

EJQTDE, 2010 No. 60, p. 17

Hence, since dim_{B} Γ_{j}(u_{0}) = 4/3 for all j, see Theorem 11(a) below, it has sense to consider M^{d}(Γ_{j}(u_{0})) for d = 4/3 only.

The sequence (M^{d}(Γ_{j}(u_{0})))_{j} will be called the Minkowski sequence associated to the trajectory Γ(u_{0}). More generally, for any trajectory Γ(u_{0}) in L^{2}(Ω) the corresponding Minkowski sequence is (M^{d_{j}}(Γ_{j}(u_{0})))_{j}, where d_{j} = dim_{B} Γ_{j}(u_{0}), provided the box dimension exists for each j. The following result deals with properties of the Minkowski sequence associated to the trajectory of the NLS initial-boundary value problem (1) at the point of the bifurcation. As the cruical step in the proof of Theorem 11 we use the result of Žubrinić and Županović [26, Theorem 6]. The simpler equivalent formulation can be found in [17, Theorem 3]. Because of the completeness, we briefly recall that result here.

Let Γ be a planar spiral defined in polar coordinates by r = f(θ), where f(θ) is decreasing to zero as θ → ∞, such that f′(θ)/(θ^{−α})′ → p as θ → ∞, α ∈ (0, 1), p > 0, and |f′′(θ)| ≤ Cθ^{−α}. Then dim_{B} Γ = d, where we defined d = 2/(1 + α), and

(28)  M^{d}(Γ) = p^{d}π(πα)^{−2α/(1+α)} \frac{1 + α}{1 − α}.

The asymptotic behaviour of the Minkowski sequence (M^{d}(Γ_{j}(u_{0})))_{j} for j → ∞ we obtain using the following well known asymptotic result for the eigenvalues of −Δ, due to H. Weyl. More precisely,

(29)  λ_{j} ≃ j^{2/N}, j → ∞,

that is, there exist positive constants a and b such that for all j, a ≤ λ_{j}/j^{2/N} ≤ b, see e.g. Mikhailov [19, Section IV.1.5] or Davies [9, Theorem 6.3.1].

Theorem 11. Assume that a_{0} = 0 in (1), and γ_{1} ≠ 0.

(a) For any u_{0} ∈ L^{2}(Ω) and j such that ⟨u_{0}, φ_{j}⟩ ≠ 0 we have dim_{B} Γ_{j}(u_{0}) = 4/3. Moreover,

(30)  M^{4/3}(Γ_{j}(u_{0})) = 3π^{1/3} (\frac{λ_{j}|⟨u_{0}, φ_{j}⟩|^{2}}{|γ_{1}| ∥u_{0}∥_{L^{2}}^{2}} )^{2/3} .

In particular,

(31)  M^{4/3}(Γ_{j}(u_{0})) ≤ 3π^{1/3} (\frac{λ_{j}}{|γ_{1}|})^{2/3} ,

and equality is achieved if and only if u_{0} ∈ span {φ_{j}}, u_{0} ≠ 0. Furthermore, we have the following asymptotic behaviour

(32)  max_{u_{0}∈L_{2}(Ω)} M^{4/3}(Γ_{j}(u_{0})) ≃ j^{4/(3N)} as j → ∞.

EJQTDE, 2010 No. 60, p. 18

(b) For any u_{0} ∈ H_{0}^{1}(Ω), u_{0} ≠ 0, we have the following identity

(33)  ∑_{j=1}^{∞}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} = \frac{\sqrt{27π}}{|γ_{1}|} (\frac{∥u_{0}∥_{H_{0}^{1}}}{∥u_{0}∥_{L^{2}}} )^{2} .

In particular,

(34)  ∑_{j=1}^{∞}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} ≥ \frac{\sqrt{27π}}{|γ_{1}|} λ_{1},

and equality is achieved if and only if u_{0} ∈ span {φ_{1}}, u_{0} ≠ 0.

(c) For u_{0} ∈ H_{0}^{1}(Ω) ∩ H^{2}(Ω), u_{0} ≠ 0, besides (33) we have the following identity:

(35)  ∑_{j=1}^{∞} λ_{j}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} = \frac{\sqrt{27π}}{|γ_{1}|} (\frac{∥u_{0}∥_{H_{0}^{1}∩H^{2}}}{∥u_{0}∥_{L^{2}}} )^{2} ,

and

(36)  M^{4/3}(Γ_{j}(u_{0})) = o(j^{−4/(3N)}) as j → ∞.

In particular,

(37)  ∑_{j=1}^{∞} λ_{j}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} ≥ \frac{\sqrt{27π}}{|γ_{1}|} λ_{1}^{2},

and equality is achieved if and only if u_{0} ∈ span {φ_{1}}, u_{0} ≠ 0.

Proof. (a) We consider the case γ_{1} > 0 (for γ_{1} < 0 the proof is analogous). After eliminating variable t from the system (18) one obtains

\begin{aligned}r_{j} & = \frac{|⟨u_{0}, φ_{j}⟩|}{ρ_{0}} (2γ_{1} \frac{arg⟨u_{0}, φ_{j}⟩ − θ_{j}}{λ_{j}} + ρ_{0}^{−2})^{−1/2} \\ & = m(−θ_{j} + θ_{0})^{−1/2}, \end{aligned} (38)

where θ_{j} → −∞, m = \frac{|⟨u_{0}, φ_{j}⟩|}{ρ_{0}} \sqrt{\frac{λ_{j}}{2γ_{1}}} , and θ_{0} is a constant. The expression (38) enables

to represent the spiral Γ_{j}(u_{0}) ⊂ ℂ in polar coordinates (r_{j}, θ_{j}) in a form r_{j} = f(θ_{j}), where f(θ_{j}) = m(θ_{0}−θ_{j})^{−1/2}. Now, the expression (30) follows directly from the result of Žubrinić and Županović [26, Theorem 6], which we briefly recalled at the beginning of this Section. We use the mentioned result in our situation by taking α = 1/2. Direct calculation gives

lim_{θ_{j}→∞} \frac{f′(θ_{j})}{θ_{j}^{−1/2}} = m,

and the expression (30) follows directly from (28). Furthermore, inequality (31) follows from (30) since |⟨u_{0}, φ_{j}⟩| ≤ ∥u_{0}∥_{L^{2}}.

EJQTDE, 2010 No. 60, p. 19

The last claim follows from (31) and the Weyl asymptotic result (29) for the eigenvalues of −Δ subject to zero boundary data.

(b) Note that if ⟨u_{0}, φ_{j}⟩ = 0 then Γ_{j} = {0}, hence M^{4/3}(Γ_{j}(u_{0})) = 0. ¿From this and using (30), we obtain

(39)  ∑_{j}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} = \frac{\sqrt{27π}}{γ_{1}} ρ_{0}^{−2} ∑_{j} λ_{j}|⟨u_{0}, φ_{j}⟩|^{2}.

Due to (13) this proves (33). Inequality (34) follows from the Poincaré inequality λ_{1}∥u_{0}∥_{L^{2}}^{2} ≤ ∥u_{0}∥_{H_{0}^{1}}^{2}, and it is optimal since the constant λ_{1} is optimal.

(c) To prove (36), note that by (35) we have λ_{j}[M^{4/3}(Γ_{j}(u_{0}))]^{3/2} → 0 as j → ∞. Hence, M^{4/3}(Γ_{j}(u_{0})) = o(λ_{j}^{−2/3}) = o(j^{−4/(3N)}), where we exploited again the Weyl asymptotic result (29).

The last claim follows from inequality ∥u_{0}∥_{H_{0}^{1}∩H^{2}} ≥ λ_{1}∥u_{0}∥_{L^{2}}, which is an immediate consequence of (13). □

Remark 12. ¿From (30) we see that M^{4/3}(Γ_{j}(u_{0})) → ∞ as γ_{1} → 0 in (1), provided ⟨u_{0}, φ_{j}⟩ ≠ 0. Another interesting consequence is that for any α ≠ 0,

M^{4/3}(Γ_{j}(αu_{0})) = M^{4/3}(Γ_{j}(v_{0})).

Hence, the mapping v_{0} ↦ M^{4/3}(Γ_{j}(u_{0})) is constant along rays through the origin in L^{2}(Ω). Since Γ_{j}(−u_{0}) = −Γ_{j}(u_{0}) for each j, this mapping can be viewed as an even function defined on the unit sphere S^{1} in L^{2}(Ω).

Remark 13. For a_{0} ≠ 0 all curves Γ_{j}(u_{0}) are rectifiable near the origin, so that dim_{B} Γ_{j}(u_{0}) = 1. Moreover, M^{1}(Γ_{j}(u_{0})) is equal to the length of the curve up to a multiplicative constant independent of j and u_{0}, see Federer [12, 3.2.39. Theorem], and its length depends on the choice of t_{0} = t_{0j} > t_{min}(u_{0}). If d = dim_{B} Γ_{j}(u_{0}) > 1 (like in Theorem 11), then M^{d}(Γ_{j}(u_{0})) does not depend on the choice t_{0j} due to excision property of d-dimensional Minkowski content, see [25, Lemma 5.6(b)].

Remark 14. To see how the Minkowski sequence depends on the domain, let us denote by Ω_{ε} = εΩ the domain obtained from Ω by scaling using the factor ε > 0. To any u_{0} ∈ L^{2}(Ω) we assign u_{0ε} ∈ L^{2}(Ω_{ε}) with u_{0ε}(x) = u_{0}(x/ε), and similarly for φ_{jε}. It is easy to see that λ_{jε} = ε^{−2}λ_{j}, and from this using (30) we obtain:

M^{4/3}(Γ_{j}(u_{0ε})) = ε^{−4/3}M^{4/3}(Γ_{j}(u_{0})).

5. Box dimension of the trajectory

For calculation of the box dimension of trajectories of NLS initial-boundary value problem (1) we use the well known fact that the box dimension is invariant with the respect to the bilipshitz mappings. As the references for this important results we refer to Falconer

EJQTDE, 2010 No. 60, p. 20 [11, Corollary 2.4(b)] and Tricot [23, see p. 121]. The main result of this section is given by theorem 17 and says that at the bifurcation value a_{0} = 0 the box dimension of trajectories of (1) viewed near the origin in L^{2}(Ω) (or near the origin of ℓ_{2}(ℂ) for (3)) has a jump. More precisely, for u_{0} ≠ 0 the mapping a_{0} ↦ dim_{B} Γ_{a_{0}}(u_{0}) is discontinuous at a_{0} = 0, where Γ_{a_{0}}(u_{0}) is the trajectory of (1) corresponding to u_{0} and a_{0}.

5.1. Multiple spirals. For a given trajectory Γ(u_{0}) in L^{2}(Ω) we define its box dimension via finite-dimensional approximations:

(40)  dim_{B} Γ(u_{0}) = lim_{k→∞} dim_{B} Π_{k}(Γ(u_{0})),

where Π_{k} is the orthogonal projection of the Lebesgue space onto span{φ_{1}, …, φ_{k}}, or equivalently, from ℓ_{2}(ℂ) onto ℂ^{k}, corresponding to the first k components of ℓ_{2}(ℂ). The above limit exists due to the monotonicity property of box dimension, see Falconer [11, p. 37].

By the multiple spiral (or n-spiral) Γ_{2N} we mean a curve in ℂ^{N} = ℝ^{2N} defined by

(41)  Γ_{2N} = {(t^{−α_{1}}e^{iλ_{1}t}, …, t^{−α_{N}} e^{iλ_{N} t}) ∈ ℂ^{N}: t ≥ t_{0}},

where t_{0} > 0. The following result will be fundamental for the computation of box dimension of trajectories of problem (1). Its consequence is that box dimension of multiply oscillating trajectories in ℂ^{N} is always less then 2. The claim extends the formula of box dimension of planar spirals due to Tricot [23, p. 121] to oscillating curves in ℂ^{N}. Since the result seems to be interesting for itself, we state it in a slightly more general form.

Theorem 15. Let α_{k} > 0 and λ_{k} ≠ 0 be given numbers, k = 1, …, N. Then the corresponding multiple spiral Γ_{2N} has box dimension

(42)  dim_{B} Γ_{2N} = max \{1, \frac{2}{1 + min_{k} α_{k}} \}.

The curve Γ_{2N} is Minkowski nondegenerate if and only if min_{k} α_{k} ≠ 1. It is rectifiable if and only if min_{k} α_{k} > 1.

Proof. (a) We assume without loss of generality that α_{1} is minimal among all α_{i}. Let Γ_{2} be the curve in ℂ defined by t ↦ t^{−α_{1}}e^{iλ_{1}t}. Let us introduce the mapping F : Γ_{2} → Γ_{2N} in the following way. First, we view the curves Γ_{2} and Γ_{2N} as subsets of ℝ^{2} and ℝ^{2N} respectively, and define (writing z_{k} = x_{k} + iy_{k}):

(43)  F (x_{1}, y_{1}) = (x_{1}, y_{1}, f_{2}(x_{1}, y_{1}), g_{2}(x_{1}, y_{1}), …, f_{n}(x_{1}, y_{1}), g_{n}(x_{1}, y_{1})),

where

f_{k}(x_{1}, y_{1}) = (λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} )^{−α_{k}} cos [λ_{k}λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} ],
g_{k}(x_{1}, y_{1}) = (λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} )^{−α_{k}} sin [λ_{k}λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} ],

EJQTDE, 2010 No. 60, p. 21 for 2 ≤ k ≤ N. It is easy to check that F maps Γ_{2} bijectively onto Γ_{2N}. Let us estimate the expression

\frac{∂f_{k}}{∂x_{1}} = C_{1}( arctan \frac{y_{1}}{x_{1}} )^{−α_{k}−1} \frac{y_{1}}{x_{1}^{2} + y_{1}^{2}} cos [λ_{k}λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} ]

+C_{2}( arctan \frac{y_{1}}{x_{1}} )^{−α_{k}} sin [λ_{k}λ_{1}^{−1} arctan \frac{y_{1}}{x_{1}} ]\frac{y_{1}}{x_{1}^{2} + y_{1}^{2}}

along Γ_{2}. Here C_{1} and C_{2} are real constants. It follows:

(44)  |\frac{∂f_{k}}{∂x_{1}} | ≤ C(t^{α_{1}−α_{k}−1} + t^{α_{1}−α_{k}}) = O(t^{α_{1}−α_{k}}), t → ∞.

Since α_{1} ≤ α_{k} the function \frac{∂f_{k}}{∂x_{1}} is bounded by a constant along Γ_{2}. Similarly for \frac{∂f_{k}}{∂y_{1}} , \frac{∂g_{k}}{∂x_{1}} and \frac{∂g_{k}}{∂y_{1}} . This proves that the mapping F is Lipschitzian. It is easy to see that the inverse F^{−1} : Γ_{4} → Γ_{2} is the projection of Γ_{4} onto Γ_{2}, which is also Lipschitzian. Hence, F is a bilipschitz function, so that we may use Falconer [11, Corollary 2.4(b)] with Tricot [23, see p. 121] to obtain that:

dim_{B} Γ_{2N} = dim_{B} F (Γ_{2}) = dim_{B} Γ_{2} = max \{1, \frac{2}{1 + α_{1}} \}.

Minkowski nondegeneracy of Γ_{2N} for α_{1} ∈ (0, 1) follows from [27, Theorem 1], since Γ_{2} is Minkowski nondegenerate, and moreover Minkowski measurable, see [26, Corollary 2], and Γ_{2N} is lipeomorphic (i.e. bilipshitz equivalent) to Γ_{2}. If α_{1} ≥ 1, then dim_{B} Γ_{2N} = dim_{B} Γ_{2} = 1. For α_{1} = 1 we have M^{1}(Γ_{2N}) = ∞ since nonrectifiability of Γ_{2} implies that Γ_{2n} is also not rectifiable. □

Remark 16. The condition on α_{i} to be positive is essential. Indeed, if N = 2 and α_{1} = α_{2} = 0, then we obtain the curve on the 2-torus, and assuming that λ_{1}/λ_{2} is rational we obtain that the curve is periodic, hence its box dimension is 1. Therefore, formula (42) does not hold in this case. A more general formulation of the first part of the theorem is as follows: if α_{j} ≥ 0 for all j and the set J = {j : α_{j} > 0, λ_{j} ≠ 0} is nonempty, then

dim_{B} Γ_{2N} = max \{1, \frac{2}{1 + min\{α_{k} : k ∈ J\}}\},

and analogously for the second part.

Now, using the result given by theorem 15 we are able to show that at the bifurcation value a_{0} = 0 the box dimension of trajectories of (1) viewed near the origin in L^{2}(Ω) (or near the origin of ℓ_{2}(ℂ) for (3)) has a jump.

Theorem 17. Assume that u_{0} ≠ 0 and γ ∈ ℝ \ {0}. Let Γ(u_{0}) be the trajectory of (1) viewed in a bounded neighbourhood of the origin. If a_{0} = 0 then dim_{B} Γ(u_{0}) = 4/3. If a_{0} ≠ 0 then dim_{B} Γ(u_{0}) = 1.

EJQTDE, 2010 No. 60, p. 22 Proof. We consider the case of γ = γ_{1} + iγ_{2} with γ_{2} = 0 and γ_{1} > 0 (for γ_{1} < 0 the proof is analogous). The case when γ_{1} ≠ 0 and γ_{2} ≠ 0, see (18), can be treated with slight modifications.

The projection of solution u(t) defined by (19), viewed in ℓ_{2}(ℂ), onto its first n components, defines the following curve in ℂ^{N}:

Γ′_{2N} … t ↦ u_{N}(t) = \frac{1}{\sqrt{2γ_{1}t + ∥u_{0}∥_{L^{2}}^{−2}}} (c_{1}e^{−iλ_{1}t}, …, c_{N}e^{−iλ_{N} t})

where c_{j} = \frac{⟨u_{0},φ_{j}⟩}{∥u_{0}∥_{L^{2}}}, t ≥ t_{0} with t_{0} large enough. We assume for simplicity that c_{j} ≠ 0 for all j. It is natural to define the curve

Γ_{2N} … t ↦ v_{N}(t) = t^{−1/2}(e^{−iλ_{1}t}, …, e^{−iλ_{N} t}).

The mapping G : Γ′_{2N} → Γ_{2N} defined by G(u_{N}(t)) = v_{N}(t) for all t ≥ t_{0} > 0, is clearly bilipschitzian, hence dim_{B} Γ′_{2N} = dim_{B} Γ_{2N}. The claim follows from Theorem 15. The case when c_{j} ≠ 0 for at least one j, is treated with minor modifications of the above proof.

If a_{0} ≠ 0, then all components z_{j}(t) are rectifiable, hence any finite-dimensional projection π_{k}(Γ_{0}) is rectifiable, where π_{k} : ℓ_{2}(ℂ) → ℂ^{k} it the natural projection onto the first k components of ℓ_{2}(ℂ). This implies that dim_{B} π_{k}(Γ_{0}) = 1 for each k, and the claim follows. □

As we saw before, assuming that a_{0} = 0 and γ_{1} ≠ 0, the natural projection of any trajectory Γ(u_{0}) into j-th component, that is, into the eigenspace spanned by φ_{j}, can be viewed as a curve in the Gauss plane either of box dimension 4/3 if ⟨u_{0}, φ_{j}⟩ ≠ 0 (in this case the projection is a spiral), or zero if ⟨u_{0}, φ_{j}⟩ = 0 (in this case the projection is simply a point).

6. Applications

In this section we introduce the notion of spiral chirp. The aim is to study the box dimension of spiral chirps and associated trajectories of NLS problems with different nonlinearities. To achieve this goal, we shall use the results on box dimensions of trajectories obtained in previous sections. Finally, in Subsection 6.3 we show that a class of planar polynomial dynamical systems can be interpreted in terms of a NLS problem of nonlocal type.

6.1. Box dimension of spiral chirps. Results on box dimension of trajectories of NLS initial-boundary value problem (1) obtained in the previous section can be used in order to calculate the box dimension of some other curves, for example spiral chirps. More precisely, in the preceding subsection we have studied box dimension of projections of Γ_{0} into subspaces of the form span {φ_{1}, …, φ_{N}}, that is, on ℝ^{2N}. Now we would like to

EJQTDE, 2010 No. 60, p. 23 consider projections onto ℝ^{2N−1}. The basic model in ℝ^{3} is the curve that we call spiral chirp:

(45)  Γ = {(t^{−α} cos t, t^{−α} sin t, t^{−α} cos t) ∈ ℝ^{3} : t ≥ t_{0}},

with α ∈ (0, 1). A more general model could be (α, β)-spiral chirp:

(46)  Γ_{3} = {(t^{−α} cos(λ_{1}t), t^{−α} sin(λ_{1}t), t^{−β} cos(λ_{2}t)) ∈ ℝ^{3} : t ≥ t_{0}},

for fixed positive α, β, t_{0}, and λ_{j} ≠ 0, j = 1, 2. Of course, box dimension of Γ_{3} will remain the same if we replace the last component with t^{−β} sin λ_{2}t.

Theorem 18. Let α and β be fixed positive numbers such that either

α ≤ β, or β ≤ α < 1, or 1 < β ≤ α.

Furthermore, assume that λ_{1}, λ_{2} are nonzero real numbers, and t_{0} > 0. Then for the spiral chirp Γ_{3} defined by expression (46) we have:

(47)  dim_{B} Γ_{3} = max \{1, \frac{2}{1 + min\{α, β\}}\}.

The curve Γ_{3} is rectifiable if and only if min{α, β} > 1.

Proof. Due to the simplicity we assume that λ_{1} = λ_{2} = 1.

Firstly, we consider the case α ≤ β.

Note that we have natural projections π_{43} : Γ_{4} → Γ_{3} and π_{32} : Γ_{3} → Γ_{2}, where Γ_{2} ⊂ ℝ^{2} is the spiral defined by t ↦ z_{1}(t) = t^{−α}e^{it}, and Γ_{4} ⊂ ℝ^{4} is 2-spiral defined by t ↦ (z_{1}(t), z_{2}(t)), with z_{2}(t) = t^{−β}e^{it}. Hence, since the projections are Lipschitzian, using known box dimensions of Γ_{2} (see Tricot [23, see p. 121]) and Γ_{4} (see Theorem 15), it follows

dim_{B} Γ_{2} = dim_{B} π_{32}(Γ_{3}) ≤ dim_{B} Γ_{3} = dim_{B} π_{43}(Γ_{4}) ≤ dim_{B} Γ_{4}.

Now, the desired result follows directly since dim_{B} Γ_{2} = dim_{B} Γ_{4} = d, where we used the notation d = max{1, 2/(1 + α)}.

Next, we consider the case when β ≤ α < 1.

Since the natural projection π_{43} : Γ_{4} → Γ_{3} is Lipschitzian, it follows that dim_{B} Γ_{3} ≤ 2/(1 + β). To prove the opposite inequality, it suffices to construct a surjective Lipschitz mapping F : Γ_{3} → Γ_{2}, where Γ_{2} is described by r = φ^{−β}, since then

\frac{2}{1 + β} = dim_{B} Γ_{2} = dim_{B} F (Γ_{3}) ≤ dim_{B} Γ_{3}.

In this way let us construct an auxiliary spiral Γ′_{2} defined by

Γ′_{2} = {(t^{−a} cos t^{b}, t^{−a} sin t^{b}) ∈ ℝ^{2} : t ≥ t_{0} > 0},

EJQTDE, 2010 No. 60, p. 24 with positive parameters a and b to be chosen later. It is easy to see that the mapping F = (F_{1}, F_{2}) where

F_{1}(x, y) = ( arctan \frac{y}{x})^{−a} cos ( arctan \frac{y}{x})^{b},
F_{2}(x, y) = ( arctan \frac{y}{x})^{−a} sin ( arctan \frac{y}{x})^{b}

maps Γ_{3} onto Γ′_{2}. Now, it remains to to find a and b such that F is Lipschitzian on Γ_{3}. First, since \frac{∂}{∂x}(arctan(y/x)) = O(t^{α}), and similarly \frac{∂}{∂y}(arctan(y/x)) = O(t^{α}), when t → ∞ it follows that

\frac{∂F_{1}}{∂x} = O(t^{α−a−1}) + O(t^{α−a+b−1}),

and similarly for the remaining first order derivatives of F_{1} and F_{2}. Finally, in order to have the boundness of |\frac{∂F_{i}}{∂x} | and |\frac{∂F_{i}}{∂y} |, it suffices to impose that α−a−1 ≤ 0 and α−a+b−1 ≤ 0.

The curve Γ′_{2} is defined by r = φ^{−a/b}, and since a/b ≤ 1, we have dim_{B} Γ′_{2} = 2/(1+(a/b)), see Tricot [23, p. 121]. In order to have dim_{B} Γ′_{2} = 2/(1 + β) we choose a and b so that β = a/b. To have b − a ≤ 1 − α we impose even b − a = 1 − α. ¿From a = βb we obtain b − a = b(1 − β) = 1 − α, hence b = (1 − α)/(1 − β), and from this a = β(1 − α)/(1 − β). This choice for a and b satisfies all the requirements. Finally, the proof for the case 1 < β ≤ α goes similarly to the previous considerations.

At the end of the proof, let us consider the rectifiability in the case when min{α, β} > 1. The spiral Γ_{2} is rectifiable, as well as the graph of the chirp z_{1}(t) = t^{β} cos t^{−1} for t near zero, corresponding to the third component of Γ_{3}. In this case the curve Γ_{3} is rectifiable as well. Indeed, since z_{1}(t) has local extrema for t_{k} such that t_{k}^{−1/β} = ±\frac{π}{2} + 2kπ, then t_{k} ≃ k^{−β} as k → ∞, hence,

(48)  l(Γ_{3}) ≤ l(Γ_{2}) + 2 ∑_{k} |z_{1}(t_{k})| ≤ l(Γ_{2}) + c ∑_{k} k^{−β} < ∞,

since β > 1. This implies that dim_{B} Γ_{3} = 1. The first inequality in (48) can be justified by isometrically transforming Γ_{3} onto the chirp defined in rectangular coordinates, defined on the interval of length of the spiral Γ_{2}.

On the other side, if min{α, β} < 1 then from (47) we see that dim_{B} Γ_{3} > 1, hence, Γ_{3} is not rectifiable. In the case when min{α, β} = 1, let us consider the following two cases:

(i) If α = 1, the spiral Γ_{2} defined by z(t) = t^{−1}e^{it} is nonrectifiable. Since M^{1}(Γ_{3}) ≥ M^{1}(Γ_{2}) = ∞, the same holds for Γ_{3}. (ii) If β = 1, then the chirp Γ′_{2} is not rectifiable. Using a suitable parametrization of Γ_{3} and the fact that F (Γ_{3}) = Γ′_{2} with a Lipschitzian map F, since Γ′_{2} is nonrectifiable then also Γ_{3} is nonrectifiable. □

Finally, using theorems 15 and 18 we can easily derive the following interesting consequence about box dimension of multiple spiral chirps.

EJQTDE, 2010 No. 60, p. 25 Corollary 19. Let Γ_{2N} ⊂ ℝ^{2N}, N ≥ 1, be a multiple spiral defined by (41), with 0 < α_{k} < 1 and λ_{k} ≠ 0, k = 1, …, N. Let e_{1}, …, e_{2N} be the canonical orthonormal base in ℝ^{2N}. Then for the multiple spiral chirp Γ_{2N−1} obtained by projecting Γ_{2N} into (2N − 1)-dimensional subspace of ℝ^{2N} spanned by any 2N − 1 vectors from the base, we have

dim_{B} Γ_{2N−1} = dim_{B} Γ_{2N} = \frac{2}{1 + min_{k} α_{k}} .

6.2. Hopf bifurcation for other types of nonlinearities. Up to now we studied the initial-boundary value problem for Schrödinger equation

u_{t} = iΔu − γuV (∥u∥_{L^{2}(Ω)}), (49)

where Ω ⊂ ℝ^{N} and γ = γ_{1} + iγ_{2} ∈ ℂ. More precisely, we considered the nonlinearity V (ρ) = ρ^{2} + a_{0}, where a_{0} was a real bifurcation parameter. In this section we consider some polynomial types of nonlinearity given by the expression

(50)  V (ρ) = ρ^{2l} + a_{l−1}ρ^{2(l−1)} + ⋯ + a_{1}ρ^{2} + a_{0},

where a_{i}, i = 0, …, l − 1 are prescribed real parameters. The NLS equation (49) corresponds to the standard generic generalized Hopf bifurcation, see Takens [22], also called the standard Hopf-Takens bifurcation. The following result extends Theorem 17.

Theorem 20. Let Γ(u_{0}) be a part of trajectory of (49) near the origin with V (ρ) defined by (50), such that u_{0} ≠ 0, γ_{2} = 0 and γ_{1} ≠ 0. Assume that k := min{j : a_{j} ≠ 0} ≥ 1. Then

(51)  dim_{B} Γ(u_{0}) = \frac{4k}{2k + 1}.

Proof. The proof uses the idea of the proof of Theorem 17, together with the proof of [26, Theorem 9(b)]. Here α = 1/(2k), so that the box dimension is obtained from Theorem 15.

□

Up to now we studied trajectories with spiral components z_{j}(t) converging to zero with equal rate α > 0, that is, |z_{j}(t)| ≃ t^{−α} as t → ∞, for all j, see for example expressions (45) or (46). Next, we are interested in dynamical systems yielding solutions in which spiral components converge to zero with different rates.

A. As the simplest model we consider a polynomial system in in ℂ^{2}, defined by

(52)  ż_{j} = −iλ_{j}z_{j} − \frac{1}{4m_{j}} z_{j}(|z_{1}|^{2m_{1}} + |z_{2}|^{2m_{2}} + a_{0}), j = 1, 2,

where m_{j} are fixed positive integers and a_{0} is the real bifurcation parameter. It is easy to see that for a_{0} < 0 an attracting invariant set S(a_{0}) is born, defined by |z_{1}|^{2m_{1}} +|z_{2}|^{2m_{2}} = −a_{0}, diffeomorphic to the unit 3-sphere in ℂ^{2}. Each trajectory starting in the point (z_{10}, z_{20}) ∈ S(a_{0}) is contained in the torus |z_{10}|S^{1} × |z_{20}|S^{1} (provided that both components z_{j0} are nonzero). The invariant set S(a_{0}) is clearly equal to the union of these trajectories.

EJQTDE, 2010 No. 60, p. 26

Assume that a_{0} = 0 and let z(0) = (z_{01}, z_{02}) ∈ ℂ^{2} be a given initial value, z(0) ≠ 0. It can be shown that the unique solution z(t) = (z_{j}(t)) of system (52) is given by

z_{j}(t) = |z_{0j}|(t + 1)^{−1/(2m_{j})} exp(−iλ_{j}t + i arg z_{0j}), j = 1, 2.

Here z_{j}(t) is a spiral converging to zero with rate α_{j} = 1/(2m_{j}). Assume that both z_{01} and z_{02} are different from zero. According to Theorem 15 the box dimension of the corresponding trajectory Γ_{4} = {(z_{1}(t), z_{2}(t)) ∈ ℂ^{2} : t ≥ t_{0} > 0} is equal to

dim_{B} Γ_{4} = \frac{2}{1 + 0.5 max\{m_{1}, m_{2}\}^{−1}} . (53)

System (52) is a special case of the following Schrödinger equation:

(54)  u_{t} = iΔu − F (u),

with initial and boundary values as in (1), and

F (u) = (|⟨u, φ_{1}⟩|^{2m_{1}} + |⟨u, φ_{2}⟩|^{2m_{2}} + a_{0}) ∑_{j=1}^{2} \frac{1}{4m_{j}} ⟨u, φ_{j}⟩ φ_{j}(x),

with scalar products in L^{2}(Ω). System (52) is obtained from (54) by considering solutions of the form u(t, x) = z_{1}(t)φ_{1}(x)+z_{2}(t)φ_{2}(x), where φ_{1} and φ_{2} are defined in the subsection 1.2.

B. Various extensions of (52) and (54) are possible, like for example a cyclic polynomial system in ℝ^{2k}, which we view as a bifurcation problem in ℂ^{k}:

ż_{1} = −iλ_{1}z_{1} − \frac{1}{4m_{1}} z_{1}(|z_{1}|^{2m_{1}} + |z_{2}|^{2m_{2}} + a_{0}) ; ż_{2} = −iλ_{2}z_{2} − \frac{1}{4m_{2}} z_{2}(|z_{2}|^{2m_{2}} + |z_{3}|^{2m_{3}} + a_{0}) ; ⋮ ; ż_{k} = −iλ_{k}z_{k} − \frac{1}{4m_{k}} z_{k}(|z_{k}|^{2m_{k}} + |z_{1}|^{2m_{1}} + a_{0}). (55)

For a_{0} < 0 an invariant set S(a_{0}) is born in ℂ^{k} defined by |z_{1}|^{2m_{1}} + |z_{2}|^{2m_{2}} = |z_{2}|^{2m_{2}} + |z_{3}|^{2m_{3}} = ⋯ = |z_{k}|^{2m_{k}}+|z_{1}|^{2m_{1}} = −a_{0}, which is a subset the surface defined by ∑_{j} |z_{j}|^{2m_{j}} = −\frac{k}{2}a_{0}, diffeomorphic to the (2k − 1)-sphere in ℂ^{k}.

Here system (55) again has the form of the Schrödinger equation (54), but with

F (u) = ∑_{j=1}^{k} \frac{1}{4m_{j}} ⟨u, φ_{j}⟩ (|⟨u, φ_{j}⟩|^{2m_{j}} + |⟨u, φ_{j+1}⟩|^{2m_{j+1}} + a_{0}) φ_{j}(x),

and the summation of indices in j + 1 is taken modulo k. Here we consider solutions of the form u(t, x) = ∑_{j=1}^{k} z_{j}(t)φ_{j}(x).

EJQTDE, 2010 No. 60, p. 27

For a_{0} = 0 and any prescribed initial value z(0) = (z_{0j}) ∈ ℂ^{k} it can be shown that the unique solution is the curve Γ_{2k} = {(z_{1}(t), …, z_{k}(t)) ∈ ℂ^{k} : t ≥ 0} of (55), defined by

(56)  z_{j}(t) = |z_{0j}|(t + 1)^{−1/(2m_{j})} exp(−iλ_{j}t + i arg z_{0j}), j = 1, …, k.

Using Theorem 15 with α_{j} = 1/(2m_{j}) we obtain the following result.

Corollary 21. Let m_{j} be positive integers and a_{0} = 0. For any trajectory Γ_{2k} of (55) such that z(0) = (z_{01}, …, z_{0k}) ≠ 0 we have

dim_{B} Γ_{2k} = \frac{2}{1 + 0.5 max\{m_{j} : z_{0j} ≠ 0\}^{−1}} .

C. Now we would like to construct a NLS Cauchy problem possessing trajectories of box dimension equal to two. Let us consider the Schrödinger equation of the form (54) such that

(57) F (u) = ∑_{k=1}^{∞}(|⟨u, φ_{2k−1}⟩|^{2m_{2k−1}} + |⟨u, φ_{2k}⟩|^{2m_{2k}} + a_{0}) ∑_{j=−1}^{0} \frac{1}{4m_{2k+j}} ⟨u, φ_{2k+j}⟩φ_{j}(x),

where m_{j} is a prescribed sequence of positive integers. First, if u ∈ L^{2}(Ω) then also F (u) ∈ L^{2}(Ω). Indeed, the expression in the round brackets is bounded by a constant M = M(u, a_{0}, m_{1}, m_{2}, …) independent of j, since ⟨u, φ_{j}⟩ converges to zero. Hence,

∥F (u)∥_{L^{2}}^{2} ≤ M^{2} ∥∑_{k=1}^{∞} ∑_{j=−1}^{0} \frac{1}{4m_{2k+j}} ⟨u, φ_{2k+j}⟩φ_{j}∥_{L^{2}}^{2} ≤ \frac{M^{2}}{16} ∥u∥_{L^{2}}^{2}.

It can be shown that if u(t) = ∑_{j} z_{j}(t)φ_{j} is a solution of (54) with initial value u(0) = u_{0} ∈ L^{2}(Ω), then z_{j}(t) is uniquely determined, and

z_{j}(t) = |⟨u_{0}, φ_{j}⟩|(t + 1)^{−1/(2m_{j})} exp(iλ_{j}t + i arg⟨u_{0}, φ_{j}⟩), j ≥ 1.

Denote by Γ(u_{0}) the trajectory in L^{2}(Ω) corresponding to t ≥ 0. Using the definition of dim_{B} Γ(u_{0}) in (40) and Theorem 15 we obtain the following result.

Corollary 22. Let a_{0} = 0. Let u(t) be the solution of (54) with F (u) defined by (57), and a_{0} = 0, satisfying initial condition u(0) = u_{0} ∈ L^{2}(Ω). Then

dim_{B} Γ(u_{0}) = \frac{2}{1 + 0.5 sup\{m_{j} : ⟨u_{0}, φ_{j}⟩ ≠ 0\}^{−1}} . (58)

In particular, if m_{j} → ∞ as j → ∞, and ⟨u_{0}, φ_{j}⟩ ≠ 0 for infinitely many j’s, then dim_{B} Γ(u_{0}) = 2.

EJQTDE, 2010 No. 60, p. 28 6.3. Box dimension of trajectories of some nonintegrable Schrödinger problems. Many dynamical systems important for applications, like for example Liénard systems, weakly damped systems etc., are not explicitly solvable. However, in spite of their nonintegrability it is possible to get the information about the box dimension of corresponding trajectories, see [30] for some examples. Here, using the connection between the nonlinear system of ODE’s and the Schrödinger equation pointed out already in Section 1.2, we would like to study the box dimension of trajectories of some nonintegrable Schrödinger evolution problems. For this purpose let us consider the following polynomial planar system:

\begin{aligned}ẋ & = f(x, y) = λ_{1}y + p(x, y) \\ ẏ & = g(x, y) = −λ_{1}x + q(x, y), \end{aligned} ; (59)

where we define p(x, y) = f(x, y) − λ_{1}y and q(x, y) = g(x, y) + λ_{1}x. Here λ_{1} is the first eigenvalue of the −Δ|_{H_{0}^{1}∩H^{2}} corresponding to an open and bounded domain Ω ⊂ ℝ^{N}. An initial point in the phase space is prescribed by x(0) = x_{10} and y(0) = y_{10}. Naturally, we say that the system (59) is equivalent to (54) if there is a bijection between the corresponding solution sets. In this sense it is easy to see that the problem (59) is equivalent to Schrödinger equation (54) with F: ℂφ_{1} → ℂφ_{1} defined by

(60) [F (u)](ξ) = (p(⟨Re u, φ_{1}⟩, ⟨Im u, φ_{1}⟩) + iq(⟨Re u, φ_{1}⟩, ⟨Im u, φ_{1}⟩))φ_{1}(ξ), ξ ∈ Ω,

and with initial function u(0) = z_{01}φ_{1}, where z_{01} = x_{10} + iy_{10} ∈ ℂ. Note that the nonlinear term F (u) is of nonlocal type, i.e. F it is not defined in pointwise manner, but via scalar products, that is, by means of two integrals. The solution of (54) in this case has the form u(t, x) = z(t)φ_{1}(x), where z(t) = x(t) + iy(t), and the components satisfy (59). Note that (54) is of nonlocal type, and in general not explicitly solvable. Moreover, it is interesting that the study of any system of ODE’s can be considered as the study of a Schrödinger problem.

Proposition 23. Any Cauchy problem for the system ẋ = f(x), where x ∈ ℝ^{n} and f: ℝ^{n} → ℝ^{n} is a Lipschitz function, can be naturally interpreted as a Schrödinger problem of the form (54), with F (u) written explicitly, see (62) below.

Proof. We can assume without loss of generality that n is even, n = 2k (if n = 2k − 1, we can add a new trivial equation ẋ_{2k} = 0, x_{2k}(0) = 0). Now the system can be written in the form

(61)  \begin{aligned}ẋ_{j} & = f_{j}(x, y) \\ ẏ_{j} & = g_{j}(x, y) \end{aligned} j = 1, …, k

where x, y ∈ ℝ^{k}. Defining z_{j} = x_{j} + iy_{j} and p_{j}(x, y) = f_{j}(x, y) − λ_{j}y_{j}, q_{j}(x, y) = g_{j}(x, y) + λ_{j}x_{j}, j = 1, …, k, where λ_{j} are the first k eigenvalues of −Δ (counting the multiplicity) with zero boundary data on a given bounded open domain Ω in ℝ^{N}, then we have ż_{j} = −iλ_{j}z_{j} + (p_{j} + iq_{j}). Multiplying with eigenfunctions φ_{j} = φ_{j}(ξ) normalized in L^{2}(Ω), and

EJQTDE, 2010 No. 60, p. 29 defining u(t, ξ) = ∑_{j} z_{j}(t)φ_{j}(ξ) we easily obtain (54), where the operator F: X_{k} → X_{k}, X_{k} = span_{ℂ}{φ_{1}, …, φ_{k}} ⊆ L^{2}(Ω), is defined by

(62)  [F (u)](ξ) = [F (∑_{j=1}^{k} z_{j}φ_{j})] (ξ) = ∑_{j=1}^{k}(p_{j}(x, y) + iq_{j}(x, y)) φ_{j}(ξ), ξ ∈ Ω,

with x_{j} = Re z_{j} = ⟨Re u, φ_{j}⟩, y_{j} = Im z_{j} = ⟨Im u, φ_{j}⟩. If we prescribe the initial value (x, y)(0) = (x_{0}, y_{0}) ∈ ℝ^{2k} for the system (61), then the corresponding initial function u(0) = ∑_{j=1}^{k}(x_{0j} + iy_{0j})φ_{j} ∈ X_{k} of the Schrödinger problem (54) generates the trajectory in X_{k}. It can be identified with the trajectory in ℝ^{2k} corresponding to (61) using the natural isomorphism between X_{k} and ℝ^{2k}. □

Remark 24. Note that the operator F: X_{k} → X_{k} in the proof of Proposition 23 is nonlocal, since (62) contains integral terms x_{j} = ⟨Re u, φ_{j}⟩ = ∫_{Ω}(Re u)(ξ)φ_{j}(ξ) dξ on the right-hand side, j = 1, …, k, and similarly for y_{j}. The operator F can be naturally extended to F̃: L^{2}(Ω) → L^{2}(Ω) by defining F̃(u) = F̃ (∑_{j=1}^{∞} z_{j}φ_{j}) = ∑_{j=1}^{k}(p_{j}(x, y) + iq_{j}(x, y))φ_{j}, where u = ∑_{j=1}^{∞} z_{j}φ_{j} is the Fourier expansion of u ∈ L^{2}(Ω) with respect to the orthonormal basis (φ_{j}) of eigenfunctons of −Δ|_{H_{0}^{1}∩H^{2}}. The claim in the proposition holds (with the same proof) for nonautonomous systems of ODE’s as well. In this case the nonlinearity in (54) takes the form F (t, u).

Remark 25. Proposition 23 enables us to reformulate the second part of the 16th Hilbert problem in terms of the Schrödinger equation (54) as follows. Let F (u) be defined by (60), where p(x, y) and q(x, y) are polynomials in 2 real variables, such that the maximum of their degrees d is at least 2. Assuming that the value of d ≥ 2 is fixed, find an upper bound for the number of limit cycles in the class of Schrödinger equations (54) in terms of d. For d = 2 (that is, for the class of quadratical planar systems of ODE’s) it is even not known if the bound can be finite.

As an example, consider the following nonlocal Schrödinger evolution problem on a given bounded domain Ω in ℝ^{N}:

(63) u_{t} = iΔu + ∑_{j=k}^{m} a_{2j} (∫_{Ω}(Re u)φ_{1} dx)^{2j} + ∑_{j=k}^{m} a_{2j+1} (∫_{Ω}(Re u)φ_{1} dx)^{2j+1} ,

where 1 ≤ k ≤ m.

Theorem 26. Let Γ(u_{0}) be the trajectory in L^{2}(Ω) of the Schrödinger evolution problem (63) viewed near the zero function in L^{2}(Ω), and generated by initial condition u(0) = z_{0}φ_{1}, with z_{0} = x_{0}+iy_{0} ≠ 0, and k is a positive integer. Assume that the coefficients a_{2}, a_{4} …, a_{2m} and a_{2k+1}, a_{2k+3}, …, a_{2m+1} are real. Assume that a_{2k+1} ≠ 0, i.e. a_{2k+1} is the first nonzero

EJQTDE, 2010 No. 60, p. 30 coefficient corresponding to an odd exponent of the integral term in (63). Then

(64)  dim_{B} Γ(u_{0}) = 2 − \frac{2}{2k + 1}.

Proof. Here we have p(x, y) = ∑_{j=1}^{m} a_{2j}x^{2k+1} + ∑_{j=k}^{m} a_{2j+1}x^{2j+1} and q(x, y) = 0. Let us write u(t, ξ) = (x(t) + iy(t))φ_{1}(ξ). Then (63) with the indicated initial condition is equivalent to the following Liénard system in the plane, see Proposition 23:

\begin{aligned}ẋ & = f(x, y) = λ_{1}y + ∑_{j=1}^{m} a_{2j}x^{2k+1} + ∑_{j=k}^{m} a_{2j+1}x^{2j+1} \\ ẏ & = g(x, y) = −λ_{1}x. \end{aligned} ; (65)

The claim follows immediately from [28, Theorem 6]. □

Remark 27. Note that under the condition of a_{2k+1} ≠ 0 the Liénard problem (65) is not explicitly solvable for (x_{0}, y_{0}) ≠ (0, 0). Therefore, the corresponding Schrödinger evolution problem (63) is also not explicitly solvable for u(0) ≠ 0.

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(Received April 1, 2010)

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia