Related papers: Exact Jacobi elliptic solutions of the $abcd$-syst…
We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation…
We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique…
The slider-block Burridge-Knopoff model with the Coulomb friction law is studied as an excitable medium. It is shown that in the continuum limit the system admits solutions in the form of the self-sustained shock waves traveling with…
In this study we compute numerical traveling wave solutions to a compact version of the Zakharov equation for unidirectional deep-water waves recently derived by Dyachenko & Zakharov (2011) Furthermore, by means of an accurate Fourier-type…
We consider elliptic solutions of the semi-discrete BKP equation and derive equations of motion for their poles. The basic tool is the auxiliary linear problem for the wave function.
We establish the existence of gravity water waves by applying a mountain pass theorem to a singular perturbation of the Alt-Caffarelli functional associated with the two-dimensional water wave equations. Our approach is formulated entirely…
We construct global curves of rotational traveling wave solutions to the $2D$ water wave equations on a compact domain. The real analytic interface is subject to surface tension, while gravitational effects are ignored. In contrast to the…
We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.
In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points,are…
Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the…
Exact solutions are obtained in the quadratic theory of gravity with a scalar field for wave-like models of space-time with spatial homogeneity symmetry and allowing the integration of the equations of motion of test particles in the…
For a five-parameter discrete $\phi^4$ model, we derive various exact static solutions, including the staggered ones, in the form of the basic Jacobi elliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of their hyperbolic…
In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially…
In this paper, we present a computer-assisted approach for constructively proving the existence of traveling wave solutions of the suspension bridge equation on the infinite strip $\Omega = \mathbb{R} \times (-d_2,d_2)$. Using a meticulous…
The Ablowitz-Ladik equation is a very important model in the nonlinear mathematical physics. In this paper, the hyperbolic function solitary wave solutions, the trigonometric function periodic wave solutions and the rational wave solutions…
In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial…
We consider in this article the system of (pure) gravity water waves in any dimension and in fluid domains with general bottoms. The unique solvability of the problem was established by Alazard-Burq-Zuily [Invent. Math, 198 (2014), no. 1,…
Using an abstract scheme of monotone semiflows, the existence of bistable traveling wave solutions of a competitive recursion system with Ricker nonlinearity is established. The traveling wave solutions formulate the strong inter-specific…
We consider axisymmetric traveling waves propagating on the gravity-driven flow of a liquid down a vertical fibre. Our starting point is the two-equation model for the flow derived in the study by Ruyer-Quil \emph{et al.} [\emph{J. Fluid…
This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical…