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Proposed in this paper is a numerical procedure to generate periodic traveling wave solutions of some nonlinear dispersive wave equations. The method is based on a suitable modification of a fixed point algorithm of Petviahvili type and…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions…
We consider an evolution equation of parabolic type in R having a travelling wave solution. We perform an appropriate change of variables which transforms the equation into a non local evolution one having a travelling wave solution with…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…
We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
It is shown that multisoliton solutions of several well known nonlinear PDEs(x, t) can be obtained by certain separation of variables: each n-soliton arises from a mutual solution of a nonlinear ODE(x), common for all NPDEs considered, and…
This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting…
Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation…
We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions,…
We discuss the behavior of solitary wave solutions of the nonlinear Schr{\"o}dinger equation (NLSE) as they interact with complex potentials, using a four parameter variational approximation based on a dissipation functional formulation of…
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
The paper takles a procedure which allow to extend some linear, wave type equations to the study of nonlinear models. More concretely, we present a practical way to generate the largest class of a given form of second order differential…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
We determine all cases when there exists a meromorphic solution of the third order ODE describing traveling waves solutions of the Kuramoto-Sivashinsky equation. It turns out that there are no other meromorphic solutions besides those…
Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from…
Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…
We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the…
In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear…