Long-time asymptotics of the good Boussinesq equation with q xx-term and its modified version

Abstract

Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with q xx-term and its modified version.

Miura transformation for the “good” Boussinesq equation

“…Remark 2. We show in Section 4 that the solution 𝑚 of RH problem 2 is unique if it exists, and that 𝑚 automatically satisfies the same symmetries (19) as the solution 𝑀 of RH problem 1; we have included (19) as part of the formulation of RH problem 1 to ensure uniqueness of the solution 𝑀.…”

“…The well‐posedness of () has been thoroughly studied (see, e.g., Refs. [9–14]) and asymptotic questions have also been investigated 15–19 …”

“…[9][10][11][12][13][14]) and asymptotic questions have also been investigated. [15][16][17][18][19] In this paper, we construct a Miura-type transformation for (1). Our main result is the discovery that each complex-valued function 𝑞 satisfying…”

Riemann-Hilbert approach and double-pole solutions for the third-order flow equation of DNLS-type equation with nonzero boundary conditions

“…In particular, for some implicit soliton solutions, it can be solved by revised Riemann-Hilbert approach [23,24]. It was also widely used in initial boundary value problem [25][26][27][28][29] and long-time asymptotics [30][31][32][33][34][35][36][37]. Due to the use of the inverse scattering transform method for solving equations often involves solving the GLM integral equation, which is not applicable for highdimensional spectral problems.…”

Time-fractional Davey–Stewartson equation: Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions

“…In the last few decades, researchers have paid much attention to the study of NPDEs, including their dynamic properties and solutions. A number of methods were put forward to acquire the solutions of NPDEs, for example, the homogeneous balance method [10][11][12][13], inverse scattering transform [14][15][16][17][18], Darboux transformation [19][20][21][22][23][24], Hirota's bilinear method [25][26][27][28][29][30][31][32][33], Riemann-Hilbert approach [34][35][36][37][38][39][40][41][42][43], nonlocal symmetry method and so on [44][45][46][47][48][49]. However, for most NPDEs, the solutions are difficult to obtain due to their complex expressions.…”