Soliton solutions of DSW and Burgers equations by generalized (G′/G)-expansion method

Abstract

The nonlinear evolution equations (NLEEs) play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Drinfeld–Sokolov–Wilson (DSW) equation and the Burgers equation are the fundamental equations occurring in various areas of physics and applied mathematics, such as nonlinear acoustics and fluid mechanics. The new generalized $(\mathrm{G{\prime}}/{\text{G}})$-expansion method is an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics. In this article, we investigate further exact solutions as well as soliton solutions to these couple of nonlinear evolution equations by executing the new generalized $(\mathrm{G{\prime}}/{\text{G}})$-expansion method. A large number of soliton solutions, including single soliton, bell-shaped soliton, kink-shaped soliton, singular kink soliton, singular soliton, periodic soliton, irregular periodic soliton solutions, and others, have been retrieved. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. We provide some 3D plots to visualize and realize the characteristics of these solutions. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations.

Appendices

Appendix A

Khan et al. (2013) used the modified simple equation method to look into exact solutions to the DSW equations and obtained 12 solutions. They obtained the following solutions

$${v}_{\mathrm{1,2}}\left(\xi \right)=\pm \omega \sqrt{\frac{6}{p(r+2s)}}tanh\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x+\omega t)\right),$$

(A.1)

$${v}_{\mathrm{3,4}}\left(\xi \right)=\pm \omega \sqrt{\frac{6}{p(r+2s)}}coth\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x+\omega t)\right),$$

(A.2)

$${v}_{\mathrm{5,6}}\left(\xi \right)=\pm I\omega \sqrt{\frac{6}{p(r+2s)}}tan\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x-\omega t)\right),$$

(A.3)

$${v}_{\mathrm{7,8}}\left(\xi \right)=\pm I\omega \sqrt{\frac{6}{p(r+2s)}}cot\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x-\omega t)\right),$$

(A.4)

$${u}_{1}\left(\xi \right)=-\frac{3\omega }{\left(r+2s\right)}{\mathit{tan}h}^{2}\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x+\omega t)\right),$$

(A.5)

$${u}_{2}\left(\xi \right)=-\frac{3\omega }{\left(r+2s\right)}{\mathit{cot}h}^{2}\left(\sqrt{\left(\frac{\omega }{2q}\right)}\left(x+\omega t\right)\right),$$

(A.6)

$${u}_{3}\left(\xi \right)=-\frac{3\omega }{\left(r+2s\right)}{\mathit{tan}}^{2}\left(\sqrt{\left(\frac{\omega }{2q}\right)}(x-\omega t)\right),$$

(A.7)

$${u}_{4}\left(\xi \right)=-\frac{3\omega }{\left(r+2s\right)}{\mathit{cot}}^{2}\left(\sqrt{\left(\frac{\omega }{2q}\right)}\left(x-\omega t\right)\right).$$

(A.8)

Appendix B

By applying the modified simple equation, Khan and Akbar (2014) investigated the Burgers equation and got five exact solutions, which are as follows:

$${u}_{1}\left(\xi \right)=-\omega \left(1-tanh\left(\frac{\omega }{4}\left(x+y-\omega t\right)\right)\right),$$

(B.1)

$${u}_{2}\left(\xi \right)=-\omega \left(1-coth\left(\frac{\omega }{4}\left(x+y-\omega t\right)\right)\right),$$

(B.2)

$${u}_{3}\left(\xi \right)=-\omega \left(1-cot\left(\frac{\omega }{4}\left(x+y-\omega t\right)\right)\right),$$

(B.3)

$${u}_{4}\left(\xi \right)=-\omega \left(1-tan\left(\frac{\omega }{4}\left(x+y-\omega t\right)\right)\right),$$

(B.4)

$${u}_{5}\left(\xi \right)=-2\omega +\frac{4\omega {c}_{1}\left(1+tanh\left(\omega \left(x+y-\omega t\right)/2\right)\right)}{2{c}_{1}\left(1+tanh\left(\omega \left(x+y-\omega t\right)/2\right)\right)+{c}_{1}\omega \left(sech\left(\omega \left(x+y-\omega t\right)/2\right)\right)}.$$

(B.5)