相关论文: Generalized Nonlinear Equation and Solutions for F…

New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions…

The magnetohydrodynamic equations system for heavy fluid over an arbitrary surface in shallow water approximation is studied in the present paper. It is shown that simple wave solutions exist only for underlying surfaces that are slopes of…

In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3 $ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a…

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters,…

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…

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider…

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…

Using Levi-Civita's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity…

We present the general formulation of the relativistic fluid dynamics with vorticity (including relativistic superfluid) on a manifold with boundary. Making use of the Hodge decomposition, we emphasize that the equations of motion include a…

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…

This paper introduces the concept of renormalized solution for a general class of non-coercive nonlinear parabolic problems, including both singularities and unbounded lower order terms. We prove existence and uniqueness of renormalized…

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…

The general Helfrich shape equation determined by minimizing the curvature free energy describes the equilibrium shapes of the axisymmetric lipid bilayer vesicles in different conditions. It is a non-linear differential equation with…

We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and generalized forces arising from a surface…

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schr\"odinger equation. The present paper contains an existence…

The boundary value problems for linear and nonlinear singular degenerate differential-operator equations are studied. We prove a well-posedeness of linear problem and optimal regularity result for the nonlinear problem which occur in fluid…

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham…

For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…

The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class…