Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval

Abstract

In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system can be expressed in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large $k$ of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length $L$ of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.

70 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1704.08561, arXiv:1704.08534

References in corpus (6)

• Integrable $\mathcal{P}T$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model
• Dark solitons in F=1 spinor Bose--Einstein condensate
• Inverse scattering transforms and N-double-pole solutions for the derivative NLS equation with zero/non-zero boundary conditions
• An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line
• The Generalized Dirichlet to Neumann map for the KdV equation on the half-line
• The mKdV equation on a finite interval

Cited by in corpus (4)

• An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line
• GHW-based Necessary and Sufficient Conditions for Support Constrained Generator Matrices
• On the Riemann-Hilbert problem for the Chen-Lee-Liu derivative nonlinear Schrödinger equation
• An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval