On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator (original) (raw)
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The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U (1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman : the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection and method of majorants.
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We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations iu t + u xx = V u ± |u| p−1 u for (x, t) ∈ R × R, in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [18] in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part v(t, x) of a solution belongs to L 2 t (0, ∞; X) for some space X. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that a local smoothing estimate of Kato type holds globally in time and combine the estimate with the Strichartz estimate to show (1 + x 2) −3/4 v L ∞ x L 2 t < ∞, which implies the asymptotic stability of a solitary wave.
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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.
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The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator; mathematically the system under study is a nonlinear Schrödinger equation, whose nonlinear term has spatial dependence of a Dirac delta function. The coupled system is invariant with respect to the phase rotation group U (1). This article, which extends the results of a previous one, provides a proof of asymptotic stability of solitary wave solutions in the case that the linearization contains a single discrete oscillatory mode satisfying a non-degeneracy assumption of the type known as the Fermi Golden Rule. This latter condition is proved to hold generically.
Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations
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Spectral stability analysis for solitary waves is developed in the context of the Hamiltonian system of coupled nonlinear Schrödinger equations. The linear eigenvalue problem for a non–self–adjoint operator is studied with two self–adjoint matrix Schrödinger operators. Sharp bounds on the number and type of unstable eigenvalues in the spectral problem are found from the inertia law for quadratic forms, associated with the two self–adjoint operators. Symmetry–breaking stability analysis is also developed with the same method.
Solitary waves and stable analysis for the quintic discrete nonlinear Schrödinger equation
Physica Scripta, 2012
The quintic discrete nonlinear Schrödinger equation (QDNLS) is an important model for describing the propagation of discrete self-trapped beams in an array of weakly coupled nonlinear optical waveguides. In this paper, the QDNLS is studied and bright solitons, dark solitons, alternating phase solitons, trigonometric function periodic wave solutions and rational wave solutions with arbitrary parameters are obtained using the extended G /G-expansion method. The linear stability of the bright soliton, the dark soliton and the rational wave solution is analyzed using the perturbation method, and the conditions that stable solitary wave solutions satisfy are presented. The stable solitary wave solutions to the QDNLS are useful in understanding the complicated physical phenomena described by QDNLS.