Related papers: Variable-coefficient symbolic computation approach…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
In this paper, nonlocal symmetries and exact solutions of variable coefficient Korteweg-de Vries (KdV) equation are studied for the first time. Using pseudo-potential, high order nonlocal symmetries of time-dependent coefficient KdV…
In this paper, we study the nonlinear dispersive waves including the rarefaction and dispersive shock waves in the discrete modified KdV equation through the numerical simulations of the dispersive Riemann problems. In particular, we…
In this article, we introduce a new class of parabolic-type pseudo differential equations with variable coefficients over the p-adics. We establish the existence and uniqueness of solutions for the Cauchy problem associated with these…
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…
In this work, we focus on the construction of Nth-rouge wave solutions for the Hirota equation by utilizing the bilinear method. The formula can be represented in terms of determinants. In addition, some interesting dynamic patterns of…
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few…
In this paper, we propose the existence and discuss the properties of rogue quantum gravitational waves. More specifically, we numerically solve the Schr\"odinger-Newton system of equations using a spectral scheme with a $4^{th}$ order…
In this paper, general higher-order rogue wave solutions of the parity-time ($\mathcal {P}\mathcal {T}$) symmetric scalar and coupled nonlocal nonlinear Schr\"{o}dinger equations (NLSEs) are calculated theoretically via a Darboux…
Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables (one per particle). In this paper we contrast them with solutions of wave equations on a space-time with multiple timelike…
We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula. We consider two types of double periodic solutions as seed solutions. We identify the squared…
A flexible and efficient method for fully vectorial modal analysis of 3D dielectric optical waveguides with arbitrary 2D cross-sections is proposed. The technique is based on expansion of each modal component in some a priori defined…
We report rogue-wave and lump patterns associated with Umemura polynomials, which arise in rational solutions of the third Painlev\'{e} equation. We first show that in many integrable equations such as the nonlinear Schr\"odinger equation…
New manifestly gauge-invariant forms of two-dimensional generalized dispersive long-wave and Nizhnik-Veselov-Novikov systems of integrable nonlinear equations are presented. It is shown how in different gauges from such forms famous…
As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic…
We find that diverse nonlinear waves, such as soliton, Akhmediev breather, and rogue waves (RWs), can emerge and interplay with each other in a two-mode coupled system. It provides a good platform to study interaction between different…
Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using parabolic front initial data. Employing a front tracking type ansatz exactly reduces the study…
The subject of this paper is multigrid solvers for Helmholtz operators with large wave numbers. Algorithms presented here are variations of the wave-ray solver which is modified to allow efficient solutions for operators with constant,…
We construct spatiotemporal localized envelope solutions of a (3+1)-dimensional nonlinear Schr\"{o}dinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique.…
General rogue waves in the Davey-Stewartson-I equation are derived by the bilinear method. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then…