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On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation. (English) Zbl 1287.35078
Authors’ abstract: We consider a system of coupled cubic nonlinear Schrödinger(NLS) equations $\text{i}\partial_t \psi_j=-\partial_x^2 \psi_j+\psi_j\sum_{k=1}^n\alpha_{jk}|\psi_k|^2, j=1,2,\dots,n,$ where the interaction coefficients $$\alpha_{jk}$$ 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 N. V. Nguyen et al. [J. Math. Phys. 54, No. 1, 011503, 19 p. (2013; Zbl 1286.35230)] and others. Specifically, we present closed-form solitary wave solutions with trivial and nontrivial-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 knowlegde, 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 $$\psi_j$$ are proportional to sech.
##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
solitary wave; nonlinear Schrodinger equation
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