Related papers: Wave Solutions
We show wave breaking for the Whitham equation in a range of fractional dispersion, i.e. the solution remains bounded but its slope becomes unbounded in finite time, provided that the initial datum is sufficiently steep.
This paper reports results on the classification of traveling wave solutions, including nonnegative weak sense, in the spatial 1D degenerate parabolic equation. These are obtained through dynamical systems theory and geometric approaches…
Quantum-mechanical wave equation for a particle with spin 1 is investigated in presence of external magnetic field in spaces with non-Euclidean geometry with constant positive curvature. Separation of the variable is performed; differential…
We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution.
This paper deals with a nonhomogeneous scalar parabolic equation with possibly degenerate diffusion term; the process has only one stationary state. The equation can be interpreted as modeling collective movements (crowd dynamics, for…
In recent time, by working in a plane with the metric associated with wave equation (the Special Relativity non-definite quadratic form), a complete formalization of space-time trigonometry and a Cauchy-like integral formula have been…
The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the…
In this paper, we study the global conservative weak solutions for a class of nonlinear dispersive wave equations after wave breaking. We first transform the equations into an equivalent semi-linear system by introducing new variables. We…
Classical wave equation is generalized for the case of viscoelastic materials obeying fractional Zener model instead of Hooke's law. Cauchy problem for such an equation is studied: existence and uniqueness of the fundamental solution is…
The goal of this work is to determine classes of traveling solitary wave solutions for Lattice Boltzmann schemes by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of…
We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher…
Due to the space and time dependence of the wave function in the time dependent Schroedinger equation, different boundary conditions are possible. The equation is usually solved as an ``initial value problem'', by fixing the value of the…
Neural field equations offer a continuous description of the dynamics of large populations of synaptically coupled neurons. This makes them a convenient tool to describe various neural processes, such as working memory, motion perception,…
We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with…
Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface…
In this paper, we overview the recent progresses on the lifespan estimates of classical solutions of the initial value problems for nonlinear wave equations in one space dimension. There are mainly two directions of the developments on the…
Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a…
Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile,…
We study the behavior of solitary-wave solutions of some generalized nonlinear Schr\"odinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…