Related papers: A Regularized System for the Nonlinear Variational…
In this paper, we study the global existence and regularity of H\"older continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a…
This paper is concerned with the final value problem for a system of nonlinear wave equations. The main issue is to solve the problem for the case where the nonlinearity is of a long range type. By assuming that the solution is spherically…
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and…
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 study a damped semi-linear wave equation in a bounded domain with smooth boundary. It is proved that any sufficiently smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset…
Based on the known implicit solution for nonlinear plasma waves, an explicit solution was obtained in the form of decomposition into harmonics. The solution obtained exhibits a mechanism for steepening of nonlinear plasma wave as a result…
The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane,…
Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for…
In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove…
We study localized solutions for the nonlinear graph wave equation on finite arbitrary networks. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest…
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…
In this short note, we derive a system of two nonlocal equations for the water-wave problem following the work of [AFM06]. Specifically, we consider a fluid with a one-dimensional free surface for an irrotational fluid both with, and…
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
A method for solving a quasilinear nonelliptical equation of the second order is developed and we give classification and parametrization of simple elements of the equation.We find exact solutions of an equation for potential stationary…
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it…
In the present paper we consider a general family of two dimensional wave equations which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated…
We prove the existence of nonlinear normal modes for general systems of two coupled nonlinear oscillators. Facilitating the comparison principle for ordinary differential equations it is shown that there exist exact solutions representing a…
This article is dedicated to the proof of the existence of classical solutions for a class of non-linear integral variational problems. Those problems are involved in nonlocal image and signal processing.
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…