Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system (original) (raw)

Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Discrete and Continuous Dynamical Systems, 2015

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system i∂tu j + ∂xxu j + 3 k=1 a kj |u k | p |u j | p−2 u j = 0, j = 1, 2, 3, where u j are complex-valued functions of (x, t) ∈ R 2 and a kj are positive constants satisfying a kj = a jk (symmetric attractive case). Our approach improves many of the previously known results. In all variational methods used previously to study the stability of solitary waves, which we are aware of, the constraint functionals were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent L 2 mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.

Solitary waves for some nonlinear Schrödinger systems

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2008

In this paper we study the existence of radially symmetric positive solutions in H 1 rad (R N) × H 1 rad (R N) of the elliptic system: − u + u − αu 2 + βv 2 u = 0, − v + ω 2 v − βu 2 + γ v 2 v = 0, N = 1, 2, 3, where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form (φ, 0) and (0, ψ) where φ and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters α, β, γ , ω has been studied recently by several authors [A.

Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities

Advances in Nonlinear Analysis, 2015

This paper proves existence and stability of solitary-wave solutions of a system of 2-coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of vector solitary-wave solutions (i.e., both components are nonzero) is established via variational methods. The set of minimizers is shown to be stable and further information about the structures of this set are given. The results extend stability results previously obtained by Cipolatti and Zumpichiatti [Nonlinear Anal. 42 (2000), 445–461], Nguyen and Wang [Adv. Differential Equations 16 (2011), no. 9–10, 977–1000; `Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system', preprint (2013)], and Ohta [Nonlinear Anal. 26 (1996), 933–939].

Existence and stability of standing waves for a coupled nonlinear schrödinger system

Acta Mathematica Scientia, 2015

We study the existence and stability of the standing waves of two coupled Schrödinger equations with potentials |x| b i (bi ∈ R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schrödinger system by solving a L 2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

Journal of Physics A: Mathematical and Theoretical, 2013

We consider a system of coupled cubic nonlinear Schrödinger (NLS) equations i ∂ψ j ∂t = − ∂ 2 ψ j ∂x 2 + ψ j n k=1 α jk |ψ k | 2 j = 1, 2,. .. , n, where the interaction coefficients α jk are real. The spectral stability of solitary wave solutions (both bright and dark) is examined both analytically and numerically. Our results build on preceding work by Nguyen et al. and others. Specifically, we present closed-form solitary wave solutions with trivial and non-trivial phase profiles. Their spectral stability is examined analytically by determining the locus of their essential spectrum. Their full stability spectrum is computed numerically using a large-period limit of Hill's method. We find that all nontrivial-phase solutions are unstable while some trivial-phase solutions are spectrally stable. To our knowledge, this paper presents the first investigation of the stability of the solitary waves of the coupled cubic NLS equation without the restriction that all components ψ j are proportional to sech.

Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system

arXiv: Analysis of PDEs, 2012

We prove existence and stability results for a two-parameter family of solitary-wave solutions to a system in which an equation of nonlinear Schr\"odinger type is coupled to an equation of Korteweg-de Vries type. Such systems model interactions between short and long dispersive waves. The results extend earlier results of Angulo, Albert and Angulo, and Chen. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we establish the sub-additivity of the problem with respect to both constraint variables jointly.

Multi-speed solitary wave solutions for nonlinear Schrödinger systems.

Journal of the London Mathematical Society, 2014

We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call multi-speeds solitary waves, behave at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backward in time and using energy methods to obtain uniform estimates.

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

Journal of Dynamics and Differential Equations

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed L 2-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger-Korteweg-de Vries systems.

Multi-Speeds solitary waves solutions for nonlinear Schrödinger systems

2012

We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call multi-speeds solitary waves, are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.