Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems with their stability

Figures & data

Figure 1. Appropriate parameter values to result (19(19) $\begin{array}{rl}{u}_1(\xi )& =a_0+\frac{\begin{array}{c}6\beta {\delta }^2\sigma (A_2\sinh \left(\xi \sqrt{-\sigma }\right)\\ +A_1\cosh \left(\xi \sqrt{-\sigma }\right)){}^2\end{array}}{\begin{array}{c}\lambda (A_1\sinh \left(\xi \sqrt{-\sigma }\right)\\ +A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2\end{array}},\\ v_1(\xi )& =\frac{\begin{array}{c}\beta {\delta }^2(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )\\ (\frac{a_0}{\beta {\delta }^2}+\frac{6\sigma (A_2\sinh \left(\xi \sqrt{-\sigma }\right)+A_1\cosh \left(\xi \sqrt{-\sigma }\right)){}^2}{\lambda (A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2})\end{array}}{2\gamma },\\ w_1(\xi )& =\frac{\begin{array}{c}\beta {\delta }^2(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )\\ (\frac{a_0}{\beta {\delta }^2}+\frac{6\sigma (A_2\sinh \left(\xi \sqrt{-\sigma }\right)+A_1\cosh \left(\xi \sqrt{-\sigma }\right)){}^2}{\lambda (A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2})\end{array}}{2\gamma }.\end{array}$(19) ) are illustrated as follows: Figures (A,B,E) describe the bright soliton and their 2D contour plot figures (B,D,F), respectively, at $\delta =1,\gamma =0.5,\beta =1.5,A_1=1.5,A_2=0.1,\sigma =-1,\lambda =1,\alpha =0,a_0=1$.

Figure 2. Appropriate parameter values to result (20(20) $\begin{array}{rl}{u}_2(\xi )& =\frac{6\beta {\delta }^2\sigma (A_1^2-A_2^2)}{\lambda (A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2},\\ v_2(\xi )& =\frac{3\mathrm{\sigma \beta }{\delta }^2(A_1^2-A_2^2)(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )}{\mathrm{\gamma \lambda }(A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2},\\ w_2(\xi )& =\frac{3\mathrm{\sigma \beta }{\delta }^2(A_1^2-A_2^2)(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )}{\mathrm{\gamma \lambda }(A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2}.\end{array}$(20) ) are depicted as follows: Figures (A,C,E) are multi-peak solitons with various amplitudes and their 2D contourplots in figures (B,D,F) respectively, at $\delta =1,\gamma =0.75,\beta =0.5,A_1=0.5,A_2=0.1,\sigma =-0.5,\lambda =1$.

Figure 3. Appropriate parameter values to the result (21(21) $\begin{array}{rl}{u}_3(\xi )& =a_0-\frac{\begin{array}{c}6\beta {\delta }^2\sigma (A_1\cos \left(\xi \sqrt{\sigma }\right)\\ -A_2\sin \left(\xi \sqrt{\sigma }\right)){}^2\end{array}}{\lambda (A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2},\\ v_3(\xi )& =\frac{\begin{array}{c}\beta {\delta }^2(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )\\ (\frac{a_0}{\beta {\delta }^2}-\frac{6\sigma (A_1\cos \left(\xi \sqrt{\sigma }\right)-A_2\sin \left(\xi \sqrt{\sigma }\right)){}^2}{\lambda (A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2})\end{array}}{2\gamma },\\ w_3(\xi )& =\frac{\begin{array}{c}\beta {\delta }^2(\sqrt{{\alpha }^2+4\mathrm{\gamma \lambda }}-\alpha )\\ (\frac{a_0}{\beta {\delta }^2}-\frac{6\sigma (A_1\cos \left(\xi \sqrt{\sigma }\right)-A_2\sin \left(\xi \sqrt{\sigma }\right)){}^2}{\lambda (A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2})\end{array}}{2\gamma }.\end{array}$(21) ) are depicted as follows: Figures (A,C,E) are periodic solitons with various amplitudes and their 2D contourplots in figures (B,D,F), respectively, at $\delta =1,\gamma =2,\beta =3,A_1=1.5,A_2=0.1,\sigma =1,\lambda =1,\alpha =0,a_0=1$.

Figure 4. Appropriate parameter values to the result (30(30) $\begin{array}{rl}{u}_2(\xi )& =\frac{2\mathrm{\beta \sigma }({\delta }^2-{\nu }^2)(2A_2{A}_1\sinh \left(2\xi \sqrt{-\sigma }\right)+A_1^2(\cosh \left(2\xi \sqrt{-\sigma }\right)+2)+A_2^2(\cosh \left(2\xi \sqrt{-\sigma }\right)-2))}{\lambda (A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2},\\ v_2(\xi )& =-\frac{\sigma (2A_2{A}_1\sinh \left(2\xi \sqrt{-\sigma }\right)+A_1^2(\cosh \left(2\xi \sqrt{-\sigma }\right)+2)+A_2^2(\cosh \left(2\xi \sqrt{-\sigma }\right)-2))(\sqrt{{\beta }^2{({\delta }^2-{\nu }^2)}^2({\alpha }^2+4\mathrm{\gamma \lambda })}+\mathrm{\alpha \beta }({\delta }^2-{\nu }^2))}{\mathrm{\gamma \lambda }(A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2},\\ w_2(\xi )& =-\frac{\sigma (2A_2{A}_1\sinh \left(2\xi \sqrt{-\sigma }\right)+A_1^2(\cosh \left(2\xi \sqrt{-\sigma }\right)+2)+A_2^2(\cosh \left(2\xi \sqrt{-\sigma }\right)-2))(\sqrt{{\beta }^2{({\delta }^2-{\nu }^2)}^2({\alpha }^2+4\mathrm{\gamma \lambda })}+\mathrm{\alpha \beta }({\delta }^2-{\nu }^2))}{\mathrm{\gamma \lambda }(A_1\sinh \left(\xi \sqrt{-\sigma }\right)+A_2\cosh \left(\xi \sqrt{-\sigma }\right)){}^2}.\end{array}$(30) ) are followed as follows: Figures (A,C,E) are the solitons of multi-peak and their 2D contourplot in figures (B,D,F), respectively, at $\delta =0.3,\gamma =0.8,\beta =1,A_1=0.5,A_2=0.1,\sigma =-0.1,\lambda =1,\alpha =0,a_0=1,y=0.1,\nu =1$.

Figure 5. Appropriate parameter values to result (32(32) $\begin{array}{r}\begin{array}{rl}{u}_4(\xi )& =\frac{2\beta ({\nu }^2-{\delta }^2)(\frac{3\sigma (A_1\cos \left(\xi \sqrt{\sigma }\right)-A_2\sin \left(\xi \sqrt{\sigma }\right)){}^2}{(A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2}+\sigma )}{\lambda },\\ v_4(\xi )& =\frac{\sigma (-2A_2{A}_1\sin \left(2\xi \sqrt{\sigma }\right)+A_1^2(\cos \left(2\xi \sqrt{\sigma }\right)+2)-A_2^2(\cos \left(2\xi \sqrt{\sigma }\right)-2))(\sqrt{{\beta }^2{({\delta }^2-{\nu }^2)}^2({\alpha }^2+4\mathrm{\gamma \lambda })}+\mathrm{\alpha \beta }({\delta }^2-{\nu }^2))}{\mathrm{\gamma \lambda }(A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2},\\ w_4(\xi )& =\frac{\sigma (-2A_2{A}_1\sin \left(2\xi \sqrt{\sigma }\right)+A_1^2(\cos \left(2\xi \sqrt{\sigma }\right)+2)-A_2^2(\cos \left(2\xi \sqrt{\sigma }\right)-2))(\sqrt{{\beta }^2{({\delta }^2-{\nu }^2)}^2({\alpha }^2+4\mathrm{\gamma \lambda })}+\mathrm{\alpha \beta }({\delta }^2-{\nu }^2))}{\mathrm{\gamma \lambda }(A_1\sin \left(\xi \sqrt{\sigma }\right)+A_2\cos \left(\xi \sqrt{\sigma }\right)){}^2}.\end{array}\end{array}$(32) ) are illustrated as follows: Figures (A,C,E) are periodic solitons with dissimilar amplitude and their 2D Contourplots in figures (B,D,F) respectively, at $\delta =0.3,\gamma =0.8,\beta =1,A_1=0.5,A_2=0.1,\sigma =-0.1,\lambda =1,a_0=1,y=0.1,\nu =1$.

Figure 6. The DR between frequency(ω) and wave number (k) of (38(38) $\begin{array}{rl}\omega & =\frac{1}{2}(2\mathrm{\delta Pϵ}-2\beta k^3-\mathrm{\alpha k}\sqrt{P}-\mathrm{\lambda k}\sqrt{P}\\ & \pm \sqrt{k^{2}P({\alpha }^2+2\mathrm{\alpha \lambda }+8\mathrm{\gamma \lambda }+{\lambda }^2)}).\end{array}$(38) ) is shown in (A) and DR between frequency(ν) and wave numbers (${\omega }_1,{\omega }_2$) of (43(43) $\begin{array}{rl}\nu & =\frac{1}{2}(-2\beta {\sigma }_1^3-2\beta {\sigma }_2^2{\sigma }_1-\alpha \sqrt{P}{\sigma }_1+2\mathrm{\delta Pϵ}-\lambda \sqrt{P}{\sigma }_1\\ & \pm \sqrt{{\alpha }^{2}P{\sigma }_1^2+2\mathrm{\alpha \lambda P}{\sigma }_1^2+8\mathrm{\gamma \lambda P}{\sigma }_1^2+{\lambda }^{2}P{\sigma }_1^2}).\end{array}$(43) ) is shown in (B).