Abstract
This paper studies the D(m, n) equation, which is the generalized version of the Drinfeld-Sokolov equation. The traveling wave hypothesis and exp-function method are applied to integrate this equation. The mapping method and the Weierstrass elliptic function method also display an additional set of solutions. The kink, soliton, shock waves, singular soliton solution, cnoidal and snoidal wave solutions are all obtained by these varieties of integration tools. Mathematics Subject Classification 37K10 · 35Q51 · 35Q55
Highlights
The theory of nonlinear evolution equations (NLEEs) has made a lot of advances in the past few decades [1,2,3,4,5,6,7,8,9,10,11,12,13]
We solve the coupled Eqs. (1) and (2) for m = 1, n = 1 by mapping methods and Weierstrass elliptic function (WEF) method [5,6,7,8] which generate a variety of periodic wave solutions (PWSs) in terms of squared Jacobi elliptic functions (JEFs) and we subsequently derive their infinite period counterparts in terms of hyperbolic functions which are solitary wave solutions (SWSs), shock wave solutions or singular solutions
This paper addressed the D(m, n) equation by the aid of a few integration tools
Summary
The theory of nonlinear evolution equations (NLEEs) has made a lot of advances in the past few decades [1,2,3,4,5,6,7,8,9,10,11,12,13]. These advances turned out to be a blessing in theoretical physics and engineering sciences where these NLEEs appear on a daily basis They appear in the study of shallow water waves in beaches and lake shores, nonlinear fiber optic communications, Langmuir and Alfven waves in plasmas, and chiral solitons in nuclear physics, just to name a few. One of the NLEEs that is very commonly studied is the Korteweg–de Vries (KdV) equation, which is studied in the context of shallow water waves on lakes and canals This equation was generalized to the K (m, n) equation about a decade ago. The most important part of the history of the D(m, n) equation is when it was solved earlier by the ansatz method where a soliton solution was obtained [3] This equation was studied with generalized evolution in the same paper during 2011 [3]. The limiting cases of these solutions, namely the topological solitons, known as kinks or shock waves, and singular solitons will be given
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