Riemann–Hilbert method and soliton dynamics for a mixed spectral complex mKdV equation with time-varying coefficients

Abstract

This paper reports the feasibility of using Riemann-Hilbert method (RHM) to reveal the soliton dynamics with variable velocity and amplitude in mixed spectral nonlinear integrable systems. To achieve this goal, we consider a novel mixed spectral complex mKdV equation (mscmKdV) with time-varying coefficients. Firstly, Lax representation of the timevarying coefficient mscmKdV equation is first abstracted from Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem equipped with a mixed spectrum. With the help of the Lax representatoin, a Riemann-Hilbert problem (RHP) associating with the mscmKdV equation is then established. Due to the confirmation of solvability of the RHP, scattering data used for recostructing the potential are further determined, and N-soliton solution of the mscmKdV equation is finally obtained. In addition, the one-and two-soliton solutions are shown to explore the characteristics of soliton dynamics in the mscmKdV equation. It is revealed fromthe mscmKdV equation that compared with the constant velocity and amplitude propagation of a soliton in the background of constant-coefficient isospectral model, the one of variablecoefficient isospectral system propagates with variable velocity but constant amplitude, while for the case of non-spectrum, whether its coefficients are variable or not, soliton always propagates with variable velocity and amplitude.

Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations

“…Of course, the coefficient functions not only cause periodicity but also affect velocity. Similar nonlinear characteristics of solitons have also been reported in [37], but the difference is that the model dealt with in this article is a three-component coupled system, which leads to computational differences and the earliest extension of the RH method [1] to the mixed spectral multi-component coupled equations with variable coefficients. There are various studies [57][58][59] on the effect of potentials on N-soliton solutions of nonlinear…”

“…However, the models solved by the RH method are basically isospectral, such as [27][28][29][30][31]. There is little research on the RH method for variable-coefficient equations [32][33][34][35], and even fewer cases [36,37] of non-isospectral or mixed spectral equations. Throughout all the existing literature, we know that Chen, Zhang, and Ye [36] successfully solved a non-isospectral Gross-Pitaevskii equation in 2021 by extending the RH method, followed by Zhang and Zhou [37] solving the variable-coefficient mixed spectral complex mKdV equation in 2023.…”

“…There is little research on the RH method for variable-coefficient equations [32][33][34][35], and even fewer cases [36,37] of non-isospectral or mixed spectral equations. Throughout all the existing literature, we know that Chen, Zhang, and Ye [36] successfully solved a non-isospectral Gross-Pitaevskii equation in 2021 by extending the RH method, followed by Zhang and Zhou [37] solving the variable-coefficient mixed spectral complex mKdV equation in 2023. For all the isospectral equations with variable coefficients that have been studied by RH method, specifically, Li, Tian, Zhang, and Yang [32] obtained multi-soliton solutions containing distinct poles of arbitrary order to the fifth-order nonlinear Schrödinger equation (NLS); Xu and Zhang [33] derived N-soliton solutions of the generalized NLS equation; Zhou and Chen [34] gained high-order soliton solutions and analyzed their long-term asymptotic properties of the inhomogeneous Hirota equation; Ma, Li, Wang, Xie, and Du [35] constructed multi-soliton solutions and gave the corresponding asymptotic analysis for the coupled Lakshmanan-Porsezian-Daniel equations.…”

“…Besides, α(t), β(t), γ(t), and δ(t) are real valued functions. In the case of u 2 = 0 and u 3 = 0, Equations ( 1)-(3) degenerate to the complex mKdV equation [37]:…”