Direct and inverse scattering problems of the modified Sawada–Kotera equation: Riemann–Hilbert approach

Abstract

It is known that both the Sawada–Kotera equation and the Kaup–Kupershmidt equation are related with the same modified equation by different Miura transformations. There is singularity at the origin in the spectral problems of the Sawada–Kotera equation and the Kaup–Kupershmidt equation. Instead, this work investigates the forward and inverse scattering problems of the modified Sawada–Kotera equation by Riemann–Hilbert approach to avoid the singularity at the origin. The Riemann–Hilbert problem along with the reconstructing formula of the modified Sawada–Kotera equation are proposed. Moreover, the properties of the reflection coefficients are analysed rigorously. The results in this paper make an important step toward the long-time asymptotics of the modified Sawada–Kotera equation.

N‐fold Darboux transformation of the discrete PT‐symmetric nonlinear Schrödinger equation and new soliton solutions over the nonzero background

“…Over the past decades, the integrable nonlinear evolution equations (NEEs) appear in diverse fields of applied mathematics and theoretical physics, 1,2 and play an important role in the description of nonlinear wave phenomena. [3][4][5] However, most of the integrable NEEs are the local equations, [6][7][8][9] that is, the solution evolution dynamics just depends on the function values at some specific time and space and the derivative values at the same point. In 2013, Ablowitz and Musslimani first proposed the following nonlocal nonlinear Schrödinger (NLS) equation: 10 i𝑞 𝑡 (𝑥, 𝑡) = 𝑞 𝑥𝑥 (𝑥, 𝑡) − 2𝜖𝑞(𝑥, 𝑡) 2 𝑞 * (−𝑥, 𝑡) (𝜖 = ±1),…”

“…Over the past decades, the integrable nonlinear evolution equations (NEEs) appear in diverse fields of applied mathematics and theoretical physics, 1,2 and play an important role in the description of nonlinear wave phenomena 3–5 . However, most of the integrable NEEs are the local equations, 6–9 that is, the solution evolution dynamics just depends on the function values at some specific time and space and the derivative values at the same point. In 2013, Ablowitz and Musslimani first proposed the following nonlocal nonlinear Schrödinger (NLS) equation: 10 $$$$\begin{align} \mathrm{i}q_t(x,t) = q_{xx}(x,t) - 2 \epsilon q(x,t)^2 q^*(-x,t) \quad (\epsilon = \pm 1), \end{align}$$$$which incorporates the parity‐time (PT) symmetric nonlocality in the nonlinear term, where $$\epsilon =-1$$ and $$\epsilon =1$$ denote the focusing and defocusing cases, respectively (the same meaning as below).…”

Assorted optical soliton solutions of the nonlinear fractional model in optical fibers possessing beta derivative

“…In recent literature, a large class of effective methods has been introduced to examine soliton solutions for different nonlinear fractional models in the context of optical fiber communications. The available techniques are: the auxiliary equation technique [11][12][13], the unified method [14], the advanced generalized ¢ G G ( ) / -expansion approach [15][16][17][18][19], the generalized exponential rational function method [20], the Chebyshev collocation method [21], the Sardar-subequation method [22], the extended tanh-function scheme [23,24], the sine-Gordon expansion technique [25], the Bernoulli sub-equation process [26], the unified method [27], the modified simple equation (MSE) approach [28], the newly extended direct algebraic technique [29,30], the rational f x tan ( ( ))-expansion and f x exp ( ( ))-expansion techniques [31], the inverse scattering technique [32], the technique of finite difference [33], the generalized Darboux transformation (gDT) method [34], the modified Kudryashov method [35], the Deift-Zhou steepest technique [36], the method of first integral [37], the steepest-descent technique [38], the Riemann-Hilbert approach [39], etc. These techniques have significantly contributed to the investigation of fractional-order models and helped improve the understanding of the complex dynamics of many real-world systems.…”

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

“…In the past few decades, various methods have been proposed and developed to obtain exact solutions to nonlinear partial differential equations (PDEs). These include the inverse scattering transform, [1] Hirota's bilinear method, [2] Darboux transformation, [3,4] Painlevé analysis, [5] the Riemann-Hilbert approach, [6,7] etc. It is well known that symmetry analysis can be used not only for simplifying PDEs but also for obtaining their exact solutions.…”