We prove that a solution to the gravity water wave problem with constant vorticity, whose profile as well its horizontal velocity component at free surface are symmetric any instant of time, is given by traveling wave. The proof based on maximum principles and structural properties governing equations.

In this paper, we take up the question of dynamic stability of genuinely two-dimensional ‘‘generalized’’ hexagonal traveling wave patterns on the surface of a three-dimensional ideal fluid (i.e., stability of Generalized Short-Crested Wave (GSCW) solutions of the water wave problem). We restrict ourselves to a study of spectral stability which considers the linearization of the water wave opera...

This study presents mathematical aspects of wave equation considered on closed space intervals. It is shown that a solution of this equation can be represented by a certain superposition of traveling waves with null values for the amplitude and for the time derivatives of the resulting wave in the endpoints of this interval. Supplementary aspects connected with the possible existence of initial...

This paper gives a new integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: ut = ∂ 5 xu − 2 3 , ut = ∂ 5 x (u 1 3 )xx − 2(u 1 6 )x u ; uxxt + 3uxxux + uxxxu = 0. The first two are in the positive order hierarchy while the 3rd one is in the negative order hierarchy. The whole hierarchy is shown integrable through solving a key...

We consider the Fisher-KPP reaction-diffusion equation in whole space. prove that if a solution has, to main order and for all times (positive negative), same exponential decay as planar traveling wave with speed larger than minimal one at its leading edge, then it has coincide aforementioned wave.

The existence and comparison theorem of solutions is first established for the quasimonotone delayed reaction-diffusion equations on R by appealing to the theory of abstract functional differential equations. The global asymptotic stability, Liapunov stability and uniqueness of traveling wave solutions are then proved by the elementary superand sub-solution comparison and squeezing methods. Sup...

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of ( − λ)u = f and to obtain a uniform L estimate for a family of bump functions traveling to infinity.

In this article we study entire solutions for a non-local delayed lattice differential equation with monostable nonlinearity. First, based on a concavity assumption of the birth function, we establish a comparison theorem. Then, applying the comparison theorem, we show the existence and some qualitative features of entire solutions by mixing a finite number of traveling wave fronts with a spati...