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 finite-difference expression, with a general solution that is a power series expansion with coefficients satisfying a nonlinear recursion relation. In the limit of ...

The orbital stability of the peaked solitary-wave solutions for a generalization of the modified Camassa-Holm equation with both cubic and quadratic nonlinearities is investigated. The equation is a model of asymptotic shallow-water wave approximations to the incompressible Euler equations. It is also formally integrable in the sense of the existence of a Lax formulation and bi-Hamiltonian stru...

or equal to Pc > U(O)*‘, where U(O)* ’ IS the propagation speed of shallow-water waves and Ue is determined by Eq. ( 12). The solitary wave is found by solving a forced Korteweg-de Vries equation. The amplitude of the solitary wave is proportional to the upstream velocity and the downstream elevation is inversely proportional to the upstream velocity. A second branch of solutions of the forced ...

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutio...

In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the 2D water wave problem in the absence of surface tension, i.e., for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water...

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the ...

Abstract. In this paper, shallow water equations (SWE) are solved through a variety of meshless methods known as radial basis functions (RBF) methods. RBF based Meshless methods have gained much attention in recent years for both the mathematics as well as the engineering community. They have been extensively popularized owing to their flexibility, power and simplicity in solving partial differ...

‎the water wave generation by wave paddle and a freely falling rigid body are examined by using an incompressible smoothed particle hydrodynamics (isph)‎. ‎in the current isph method‎, ‎the pressure was evaluated by solving pressure poisson equation using a semi-implicit algorithm based on the projection scheme and the source term of pressure poisson equation contains both of divergence free ve...