Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

Abstract

In the first part of these notes, we deal with first order Hamiltonian systems in the form Ju (t) = ∇H(u(t)) where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian H ∈ C 1 (X, R) is required to be invariant with respect to the action of a group {e tA : t ∈ R} of isometries where A ∈ B(X, X) is skew-symmetric and JA = AJ. A standing wave is a solution having the form u(t) = e tλA ϕ for some λ ∈ R and ϕ ∈ X such that λJAϕ = ∇H(ϕ). Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves.In the second part we consider the nonlinear Schrödinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Mathematics Subject Classification (2000). Primary 37-01; Secondary 37K45, 35Q55.

Orbital Stability: Analysis Meets Geometry

“…In [Stu08], E is a Hilbert space and still a different formulation is adopted. Basically, the domain D is not introduced, the equation (6.5) is interpreted as an equation in E * and the time derivative is understood as a strong derivative for E * -valued functions.…”

“…As an example, if E = H 1 (R d , C) and depending on the problem considered, one may want to use either the L 2 inner product or the H 1 inner product: in Section 9 the first choice is made and in Section 10 the second one. In the formalism developed in [GSS87,GSS90,Stu08], E is always supposed to be a Hilbert space, and only the Hilbert space inner product is used in the analysis of the Hessian. But the introduction of a second inner product is a regularly used device in the literature on orbital stability for the Schrödinger in particular.…”

“…We can now formulate the following simple but crucial technical result, which is a multidimensional version of Lemma 2.1 in [Stu08].…”

Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrödinger equations with potential

“…[7,36]. The main reason is that due to the presence of the quasilinear term (∆|u| 2 )u, we do not know if the minimizer obtained in Theorem 4.3 is unique.…”

Standing waves in nonlinear Schrödinger equations

“…[19,23,24,40,44,46]. See also [72] for a review of the different ways to obtain stability for general nonlinearities thanks to Grillakis-Shatah-Strauss' results. Although theses notes are restricted to nonlinear Schrödinger equations with powertype nonlinearities, our goal is to provide the reader with methods applicable in rather general situations and this is why we choose to present the proof of Theorem 4.3 using the following stability criterion.…”