Initial-Boundary Value Problems for the Coupled Higher-Order Nonlinear Schrödinger Equations on the Half-line

Abstract

The unified transform method is used to analyze the initial-boundary value problem for the coupled derivative nonlinear Schrödinger(CDNLS) equations on the half-line. In this paper, we assume that the solution u(x, t) and v(x, t) of CDNLS equations are exists, and we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ.

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

“…Proposition 2.3. For each n = 1, 2, 3, 4 and ∈ D n , the function P n (x, t, ) is defined well by Equation (27). For any identified point (x, t), P n (x, t, ) is bounded and analytical as a function of ∈ D n away from a possible discrete set of singularities { j } at which the Fredholm determinant vanishes.…”

“…These equations compose the matrix decomposition problem of {s, S} by use {R n , S n , T n }. In fact, by the definitions of the integral Equation (27) and {R n , S n , T n }, we obtain…”

“…Proposition 2.7. Suppose that P n (x, t, )(n = 1, 2, 3, 4) are the eigenfunctions defined by (27), and the j (j = 1, 2 · · · L) of singularities are as the abovementioned assumptions. Then, we have the following residue conditions hold true:…”

“…The system has been derived as a model for describing the propagation of short pulses in birefringent optical fibers both in picosecond and femtosecond regions, and which is a hybrid of the coupled NLS equation and coupled derivative NLS equation, because case γ =0 the system is known as the Manakov system: And δ =0 the system is the coupled derivative NLS equation The system has been studied extensively on the integrability related to explicit form of the Lax pair and infinitely many local conservation laws, bilinear representations, soliton solutions given by the Hirota's bilinear method, and rogue waves solutions given by the Darboux transformation method . Recently, the IBV problem of the Manakov system and the IBV problem of the coupled derivative NLS equations are also studied by means of the Fokas approach, respectively. To the best of our knowledge, the IBV problem of the system has not been studied.…”

“…18,21,25,26 These authors have also done some work on integrable equations with 2 × 2 or higher order Lax pairs. 10,[27][28][29] As is well known, many important PDEs can be used to describe nonlinear physical phenomena in nature. Among them, the modified NLS equation is one of the most important mathematical physics models…”