Long-Time Asymptotics of a Three-Component Coupled mKdV System

Abstract

We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method.

Inverse scattering transforms and soliton solutions of nonlocal reverse‐space nonlinear Schrödinger hierarchies

“…We also point out that it would be particularly interesting to generate different kinds of explicit and exact solutions of integrable equations, such as positon solutions and complexiton ones, 47,48 lump and lump-type solutions, [49][50][51] solitonless solutions, 52,53 algebrogeometric solutions, 54,55 and dromions 56,57 from a perspective of Riemann-Hilbert problems. Moreover, it is worthy for further investigation is how to establish Riemann-Hilbert problems for dealing with extended integrable counterparts including super or supersymmetric equations, integrable couplings, and fractional analogous equations.…”

“…Under the conditions (45), the analytic Fredholm theory (or more precisely, the Volterra theory on integral equations) guarantees that the two eigenfunctions ± exist, and allow analytical continuations off the real line ∈ ℝ as soon as the both integrals on their right-hand sides converge. Noting the diagonal form of Λ and the first assumption in (53), one can observe that the integral equation for the last columns of + contains only the exponential factor e − ( − ) , which also decays due to > in the integral, when takes values in ℂ + , and the integral equation for the first column of − contains only the exponential factor e ( − ) , which decays due to < in the integral, if takes values in ℂ + . Thus, these + 1 columns are analytical with respect to in ℂ + and they are continuous with respect to inC + .…”

Exact solutions to the nonlinear equation in traffic congestion

“…In order to obtain the exact solutions, many useful methods have been proposed such as the nonlinear steepest descent method [17], collocation method [18], direct method [19], the tanh-sech method [20], sine-cosine method [21,22], symmetrical method [23,24], Hirota bilinear method [25,26], and so on. Nowadays, a powerful method named the complete discrimination system for polynomial method has been proposed to obtain the classification of single traveling wave solutions to a series of nonlinear differential equations [27][28][29][30][31][32].…”

New Interaction Solutions for a (2 + 1)-Dimensional Vakhnenko Equation

“…It is well known that nonlinear partial differential equations (NPDEs) and their solutions play a significant role in interpreting many important phenomena in nonlinear sciences. A variety of powerful methods are developed for finding the exact solutions of NPDEs, such as Hirota's method [1,2], simplified Hirota's method [3,4], the Lie symmetry analysis method [5,6], the simplest equation method [5,6], the invariant subspace method [7], and the nonlinear steepest descent method [8]. Very recently, the lump and interaction solutions [9][10][11] have attracted the attention of many scholars because of lump's applications in nonlinear optics, physics, oceanography, etc, and the interaction solutions are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves [12,13].…”