On nonlocal models of Kulish-Sklyanin type and generalized Fourier transforms

Abstract

A special class of multicomponent NLS equations, generalizing the vector NLS and related to the {\bf BD.I}-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a Riemann-Hilbert problem. We introduce the minimal sets of scattering data $\mathfrak{T}$ which determines uniquely the scattering matrix and the potential $Q$ of the Lax operator. The elements of $\mathfrak{T}$ can be viewed as the expansion coefficients of $Q$ over the `squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the mapping $\mathfrak{T}\to Q$ is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor ($F=1$ and $F=2$, respectively) Bose-Einstein condensates.

16 pages

References in corpus (9)

• Making Sense of Non-Hermitian Hamiltonians
• Dark solitons in F=1 spinor Bose--Einstein condensate
• PT-symmetric quantum mechanics
• Perturbation theory for bright spinor Bose--Einstein condensate solitons
• Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory
• Multicomponent Bright Solitons in F = 2 Spinor Bose-Einstein Condensates
• Dynamical symmetry in spinor Bose-Einstein condensates
• Basic aspects of soliton theory
• New Integrable Multi-Component NLS Type Equations on Symmetric Spaces: Z_4 and Z_6 Reductions

Cited by in corpus (4)

• Nonlocal Modified KdV Equations and Their Soliton Solutions
• Multi-component AKNS systems
• Soliton solutions of the shifted nonlocal NLS and MKdV equations
• Hirota bilinear forms of the AKNS($N$) systems