Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval

Abstract

In this paper, we study the initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the finite interval with Lax pairs involving 3 × 3 matrices via the Fokas method. We write the solution in terms of the solution of a 3 × 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrixvalue spectral functions s(k), S(k), and S L (k), which are determined by the initial values, boundary values at x = 0, and at x = L, respectively. Some of the boundary values are known for a well-posed problem, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.

Solitary Wave and Quasi-Periodic Wave Solutions to a (3+1)-Dimensional Generalized Calogero-Bogoyavlenskii-Schiff Equation

“…It is also integrable by the one-dimensional inverse scattering transform and Painléve test [28][29][30][31]. Recently, more and more people are interested in studying some generalized nonlinear evolution equations [32][33][34][35][36][37][38], resulting from their more widely applications in many physical fields [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. To our knowledge, Riemann theta function periodic wave solutions for Eq.…”

The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

“…In addition, Yang proposed inverse scattering transform based on RH problem. Furthermore, the RH method can also be used to study the initial boundary value problem of integrable nonlinear equations and the asymptotic behavior of solutions . It is well known that the nonlinear Schrödinger equation can be used to describe physical phenomena in deep water, optics, and plasma physics.…”

“…Furthermore, the RH method can also be used to study the initial boundary value problem of integrable nonlinear equations and the asymptotic behavior of solutions. [11][12][13][14][15][16][17][18][19][20][21] It is well known that the nonlinear Schrödinger equation can be used to describe physical phenomena in deep water, optics, and plasma physics. However, due to the existence of higher-order effects, such as the third-order dispersion (TOD), the self-steepening (SS), and the stimulated Raman scattering (SRS) effects, some higher-order nonlinear Schrödinger equations, such as two components and three components, have been proposed.…”

Characteristics of rogue waves on a soliton background in a coupled nonlinear Schrödinger equation

“…In fact, the localized wave solutions on the plane wave background for NLS equation had been found a long time ago . In addition to the NLS equation, it has been known that there exist Kuznetsov‐Ma (KM) soliton, Akhmediev breather (AB), and Peregrine soliton (PS) for the most NLS‐type equations . In recent years, based on Darboux transformation (DT) approach and Hirota's bilinear approach, there have been a number of effects to investigate soliton solutions and rogue wave solutions of nonlinear differential equations .…”

“…15 In addition to the NLS equation, it has been known that there exist Kuznetsov-Ma (KM) soliton, Akhmediev breather (AB), and Peregrine soliton (PS) for the most NLS-type equations. [16][17][18][19][20][21][22][23][24][25][26][27] In recent years, based on Darboux transformation (DT) approach and Hirota's bilinear approach, there have been a number of effects to investigate soliton solutions and rogue wave solutions of nonlinear differential equations. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] In particular, one of our authors, Wang, [48][49][50][51][52][53][54] investigates the soliton solutions and rogue wave solutions of nonlinear evolution equations (NLEEs).…”