Topological and Nontopological 1-Soliton Solution of the Generalized KP-MEW Equation

Abstract

In this paper, we obtain the topological and nontopological 1-soliton solution of the generalized Kadomtsev–Petviashvili modified equal width (KP-MEW) equation. The use of solitary wave ansatz method in context of doubly periodic Jacobi elliptic functions is done, which leads to the exact topological and nontopological soliton solutions. The Jacobi elliptic function solution degenerates into solitary wave solution in the limiting case of the modulus parameter. We derive the power law nonlinearity parameter domain for the existence of soliton solution, which is different for the topological and nontopological soliton. Also we identify the parametric restriction on the coefficients for the existence of solitary wave solutions. Finally, the remarkable features of such solitons are demonstrated in several interesting figures.

Appendix

Here we will furnish a primary introduction about elliptic functions (for more details, see [26, 27]). Three Jacobian elliptic functions are defined as

$$\begin{aligned} \mathrm{{sn}}(x,k)=\mathrm{{sin}}~\varphi , \mathrm{{cn}}(x,k)=\mathrm{{cos}}~\varphi , \qquad \mathrm{{dn}}(x,k)=\mathrm{{d}}\varphi /\mathrm{{d}}x \end{aligned}$$

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where the amplitude function $\varphi (\mathrm{{z}},k)$ is defined by the integral

$$\begin{aligned} \mathrm{{z}}(\varphi ,k)=\int _{0}^{\varphi }\frac{\mathrm{{d}}\tau }{\sqrt{1-k^2\mathrm{{sin}}^2\tau }} \end{aligned}$$

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The square of the real number k is called elliptic modulus parameter and $k^2\in (0,1)$. Also $k'^2=1-k^2$ is called complementary modulus parameter. To avoid the complexity, in the text we inhibit the explicit modular dependence and write $\mathrm{{sn}}x,\mathrm{{cn}}x,\mathrm{{dn}}x$ etc. These are doubly periodic functions of periods $4K,2iK';4K,4iK'$, and $2K,4iK'$, respectively, where the quarter-periods K and $K'$ are the real numbers given by

$$\begin{aligned} K(k)\equiv K=\mathrm{{z}}(\pi /2,k),\qquad K'(k)\equiv K'=K(k') \end{aligned}$$

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K is called complete elliptic integral of second kind. Some useful relations are

$$\begin{aligned} \mathrm{{sn}}^2x+\mathrm{{cn}}^2x=1, \mathrm{{dn}}^2x+k^2\mathrm{{sn}}^2x=1,\qquad k^2\left( \mathrm{{cn}}^2x-1\right) =\mathrm{{dn}}^2x-1 \end{aligned}$$

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the rules of differentiation are

$$\begin{aligned} \mathrm{{sn}}'x=\mathrm{{cn}}x~\mathrm{{dn}}x,\qquad \mathrm{{cn}}'x=-\mathrm{{sn}}x~\mathrm{{dn}}x, \qquad \mathrm{{dn}}'x=-k^2\mathrm{{sn}}x~\mathrm{{cn}}x \end{aligned}$$

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and

$$\begin{aligned} \mathrm{{sn}}(x,k) \xrightarrow [k\rightarrow 0]{k\rightarrow 1} \left\{ \begin{array}{ll} \mathrm{{tanh}}x \\ \mathrm{{sin}}x \end{array} \right. , \quad \mathrm{{cn}}(x,k) \xrightarrow [k\rightarrow 0]{k\rightarrow 1} \left\{ \begin{array}{ll} \mathrm{{sech}}x \\ \mathrm{{cos}}x \end{array} \right. , \quad \mathrm{{dn}}(x,k) \xrightarrow [k\rightarrow 0]{k\rightarrow 1} \left\{ \begin{array}{ll} \mathrm{{sech}}x \\ 1 \end{array} \right. \end{aligned}$$

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