# On a coupled water wave system

TL;DR: In this paper, the authors consider a coupled system which describes the propagation of dispersive water waves and demonstrate several solutions of the equation by an explicit analytical method, under certain conditions solitary wave solutions can be obtained.

Abstract: Research in recent years shows that a number of physically significant problems can be modelled by coupled nonlinear wave equations. It is observed that most of such equations exhibit solitary wave solutions which reveal the important properties of the system concerned. In this work, we consider a coupled system which describes the propagation of dispersive water waves. By an explicit analytical method we demonstrate several solutions of the equation. It is worth remarking that under certain conditions solitary wave solutions can be obtained. Finally, some interesting results are also found by introducing cubic nonlinearity in the basic system.

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TL;DR: In this article, a new family of coupled wave equations which are basically nonlinear in nature is introduced and an analytical study enables us to show that these equations exhibit solitary wave profiles.

Abstract: This paper deals with a new family of coupled wave equations which are basically nonlinear in nature. An analytical study enables us to show that these equations exhibit solitary wave profiles. Finally some remarks are drawn from the standpoint of atmospheric problem.

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TL;DR: In this paper, a coupled Korteweg-de Vries equation is presented, which exhibits a soliton solution and three basic conserved quantities for a special choice of dispersion relations.

757 citations

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TL;DR: In this article, a commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed.

Abstract: A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.

440 citations

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TL;DR: In this paper, a coupled nonlinear wave equation is presented, and it is shown that the coupled equation possesses infinitely many symmetries and conservation laws, each of which is a hamiltonian system.

179 citations

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TL;DR: A review of mesoscale coherent structures observable in the ocean and the atmosphere is presented in this paper, where the authors discuss the importance of solitary waves in large-scale geophysical motions and the numerical accuracy needed to reproduce them.

Abstract: Publisher Summary This chapter analyzes the planetary solitary waves in geophysical flows. It outlines the importance of solitary waves in large-scale geophysical motions. Large-scale motions of geophysical fluids and the related solitary wave models are studied. A review of the body of research carried out on mesoscale coherent structures observable in the ocean and the atmosphere is presented. The observation of these structures has been the primary motivation for a renewed interest in nonlinear, permanent-form waves, solutions of one-dimensional model equations endowed with remarkable properties. The chapter illustrates the one-dimensional equations, which are relevant for the dynamics of oceanic and atmospheric motions and of their solutions. A study of the initial value problem posed by the evolution of coherent structures is presented. The numerical accuracy needed to reproduce them is examined. The transition to the range of linear dispersive dynamics through the effects of dissipation is examined. Various functional laws of successively higher order and, therefore, more scale selective are also considered.

55 citations

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TL;DR: Explicit solutions of a model equation describing the motion of melts in the Earth are obtained by an independent variable transformation as discussed by the authors, including periodic wave solutions, solitary wave solutions and weak solutions with compact support.

Abstract: Explicit solutions of a model equation describing the motion of melts in the Earth are obtained by an independent variable transformation. Included are periodic wave solutions, solitary wave solutions and weak solutions with compact support. It is also discussed that if the model equation is modified, the resultant equation can be reduced to the exactly solvable Korteweg-de Vries equation.

34 citations