The Lie-group formalism is applied to investigate the symmetries of the Dullin-Gottwald-Holm equation φt −αφxxt +2wφx +3φφx + γφxxx = α(2φxφxx +φφxxx), which describes the unidirectional propagation of two dimensional waves in shallow water over a flat bottom. We derived the infinitesimals that admit the classical symmetry group. The reduced ordinary differential equation is further studied and...

We apply a multiple–time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity–free perturbation theory, we show that the well known N–soliton dynamics of the shallow water wave equation, in the particular case of α = 2β, can be reduced to the N–soliton solution that satisfies si...

In this paper, the equation of motion for an incompressible transversely isotropic fibre-reinforced elastic solid is derived in terms of a scalar function.   The general solution of the equation of motion is obtained, which satisfies the required radiation condition.  The appropriate traction free boundary conditions are also satisfied by the solution to obtain the required secular equation for...

Many dynamical problems in physics and other natural fields are usually characterized by the nonlinear evolution of partial differential equations known as governing equations. Searching for an analytical exact solution to a nonlinear system has long been an important and interesting topic in nonlinear science both for physicists and mathematicians, and various methods for obtaining exact solut...

in this paper, the coupled dispersive (2+1)-dimensional long wave equation is studied. the traveling wave hypothesis yields complexiton solutions. subsequently, the wave equation is studied with power law nonlinearity where the ansatz method is applied to yield solitary wave solutions. the constraint conditions for the existence of solitons naturally fall out of the derivation of the soliton so...

We examined the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which arises not only in fluid dynamics, superfluids, physics, and plasma physics but also construction of connections between hydrodynamic optical model fields. Moreover, unlike Kadomtsev–Petviashvili equation (KPE), KP-B allows modeling waves traveling both directions does require zero-mass assumption, is nec...

In this paper we develop a higher-order nonlinear Schrodinger equation with variable coefficients to describe how a water wave packet will deform and eventually be destroyed as it propagates shoreward from deep to shallow water. It is well-known that in the framework of the usual nonlinear Schrodinger equation, a wave packet can only exist in deep water, more precisely when kh > 1.363 where k i...

A ( $$3+1$$ )-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton solutions are constructed. In the process of degeneration solutions, T-breathers derived by taking complexication method. Then rogue will emerge during breathers parameter limit Through full N-soliton, M-lump based long-wave...