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Localization in optical systems with an intensity-dependent dispersion. (English) Zbl 1479.35820

Summary: We address the nonlinear Schrödinger equation with intensity-dependent dispersion which was recently proposed in the context of nonlinear optical systems. Contrary to the previous findings, we prove that no solitary wave solutions exist if the sign of the intensity-dependent dispersion coincides with the sign of the constant dispersion, whereas a continuous family of such solutions exists in the case of the opposite signs. The family includes two particular solutions, namely cusped and bell-shaped solitons, where the former represents the lowest energy state in the family and the latter is a limit of solitary waves in a regularized system. We further analyze the delicate analytical properties of these solitary waves such as their asymptotic behavior near singularities, the convergence of the fixed-point iterations near such solutions, and their spectral stability. The analytical theory is corroborated by means of numerical approximations.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q60 PDEs in connection with optics and electromagnetic theory
78A60 Lasers, masers, optical bistability, nonlinear optics
35P15 Estimates of eigenvalues in context of PDEs
35C08 Soliton solutions
35B65 Smoothness and regularity of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65T50 Numerical methods for discrete and fast Fourier transforms
65H10 Numerical computation of solutions to systems of equations

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References:

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