Abstract
In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model.
Highlights
The study of nonlinear structures is an attractive subject that has captured the minds of many researchers in the twentieth century due to its importance in many fields of science such as optical fiber, Ocean, water tank, physics of plasmas, quantum field theory, Bose-Einstein condensate (BEC), etc. [1,2,3,4,5,6,7,8]
Both ordinary differential equations (ODEs) and partial differential equations (PDEs) have played an effective role in explaining the mechanism and ambiguities of several phenomena that occur in nature on the visible and invisible levels [9,10,11,12,13,14,15]
Some new analytical solutions in terms of the Weierstrass elliptic double periodic function and Jacobi elliptic function to the generalized Degasperis Procesi equation and its family are constructed in an explicit form
Summary
The study of nonlinear structures is an attractive subject that has captured the minds of many researchers in the twentieth century due to its importance in many fields of science such as optical fiber, Ocean, water tank, physics of plasmas, quantum field theory, Bose-Einstein condensate (BEC), etc. [1,2,3,4,5,6,7,8]. Many equations have been devoted for modelling solitary waves such as the family of KdV equation, KadomtsevPetviashvili (KP) equation, Zakharov-Kuznetsov (ZK) equation, and Benjamin-Bona-Mahony (BBM) equation [1, 2, 21,22,23,24] All these families were used for describing the unmodulated solitons that propagate with phase velocity and with smooth crest. The gDP equation and its family are considered good mathematical models for studying the propagation of nonlinear shallow water waves (specially solitary surface wave, peakons, cuspons, periodic waves, and sometimes shock waves) with small amplitude and long wavelength. It has the following general form [31,32,33]. It should be mentioned that in the present study, we will obtain some a new Nonlinear structures of the generalized Degasperis Procesi equation and its application to plasma physics explicit form to the traveling solutions while most published papers only mentioned the conditions for the existence of these solutions and do not provide an explicit picture for these solutions
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