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Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
A novel mathematical nonlinear theory of surface gravity waves in deep water is presented, in which analytical analysis of the classical nonlinear equations of fluid dynamics is performed under less restrictive assumptions than those…
We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous…
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
This article deals with rotating magnetohydrodynamic flows of a thin stratified layer of astrophysical plasma in a gravitational field with a free-surface in a vertical external magnetic field. Magnetohydrodynamic equations are obtained in…
In this article we obtain exact solutions of (2+1)-dimensional Boiti-Leon-Pempinelli system of nonlinear partial differential equations which describes the evolution of horizontal velocity component of water waves propagating in two…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering…
The purpose of this article is to provide a perspective -- admittedly, a rather subjective one -- of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of…
The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the…
We present full classification of Q-conditional symmetries for the two-dimensional nonlinear wave equation.
The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this…
The article explores the acoustic equations in inhomogeneous media and the linearized shallow water equations. Two methods for integrating these equations are proposed. The first method is based on the of the Laplace cascade method, while…
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of…
Symmetries for wave equation with additional conditions are found. Some conditions yield infinite-dimensional symmetry algebra for the nonlinear equation. Ansatzes and solutions corresponding to the new symmetries were constructed.
Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface…
We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling…
In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use…
A novel third order nonlinear evolution equation governing the dynamics of high frequency electrostatic drift waves has been derived in the framework of a plasma fluid model in an inhomogeneous magnetized plasma. The linear dispersion…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.