The eigenvalue problem for the focusing nonlinear Schrödinger equation: New solvable cases

DOI10.1016/S0167-2789(00)00126-3zbMath0984.34072MaRDI QIDQ5940287

Alexander Tovbis, Stephanos Venakides

Publication date: 28 April 2002

Published in: Physica D (Search for Journal in Brave)

zbMATH Keywords

hypergeometric functionssemi-classical limitfocusing nonlinear Schrödinger equationZakharov-Shabat eigenvalue problem

Mathematics Subject Classification ID

Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs (35P30) Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) (34L40) NLS equations (nonlinear Schrödinger equations) (35Q55) Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems (37K15) Scattering theory, inverse scattering involving ordinary differential operators (34L25)

Related Items

THE RIEMANN–HILBERT APPROACH TO OBTAIN CRITICAL ASYMPTOTICS FOR HAMILTONIAN PERTURBATIONS OF HYPERBOLIC AND ELLIPTIC SYSTEMS ⋮ The Wentzel-Kramers-Brillouin approximation of semiclassical eigenvalues of the Zakharov–Shabat problem ⋮ The Gaussian semiclassical soliton ensemble and numerical methods for the focusing nonlinear Schrödinger equation ⋮ On the spectrum of the periodic focusing Zakharov-Shabat operator ⋮ Elliptic finite-band potentials of a non-self-adjoint Dirac operator ⋮ Global well-posedness for the derivative non-linear Schrödinger equation ⋮ The semiclassical modified nonlinear Schrödinger equation I: Modulation theory and spectral analysis ⋮ Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation ⋮ Two soliton collision for nonlinear Schrödinger equations in dimension 1 ⋮ Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data ⋮ Numerical solution of the nonlinear Schrödinger equation by feedforward neural networks ⋮ A second look at the Gaussian semiclassical soliton ensemble for the focusing nonlinear Schrödinger equation ⋮ On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation ⋮ Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems ⋮ Direct search for exact solutions to the nonlinear Schrödinger equation

Cites Work

• Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem
• On the semiclassical limit of the focusing nonlinear Schrödinger equation
• A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item