Soliton and quasi-periodic wave solutions for b-type Kadomtsev–Petviashvili equation

Abstract

In this paper, truncated Laurent expansion is used to obtain the bilinear equation of a nonlinear evolution equation. As an application of Hirota’s method, multisoliton solutions are constructed from the bilinear equation. Extending the application of Hirota’s method and employing multidimensional Riemann theta function, one and two-periodic wave solutions are also obtained in a straightforward manner. The asymptotic behavior of one and two-periodic wave solutions under small amplitude limits is presented, and their relations with soliton solutions are also demonstrated.

Appendix

$$\begin{aligned} a_{11}=&\sum _{n=-\infty }^{\infty }(-16\,{\pi }^{2}n_{{1}}n_{{2}}\alpha _{{2}}-16\,{\pi }^{2}{n_{{1}}}^{2} \alpha _{{1}})\mathfrak {I}_{1}(n) \end{aligned}$$

(45)

$$\begin{aligned} a_{12}=&\sum _{n=-\infty }^{\infty }(-16\,{\pi }^{2}n_{{2}}n_{{1}}\alpha _{{1}}-16\,{\pi }^{2}{n_{{2}}}^{2} \alpha _{{2}})\mathfrak {I}_{1}(n), a_{13}=\sum _{n=-\infty }^{\infty }\mathfrak {I}_{1}(n)\end{aligned}$$

(46)

$$\begin{aligned} a_{14}=&\sum _{n=-\infty }^{\infty }(-480\,{\pi }^{2}{n_{{1}}}^{2}{\alpha _{{1}}}^{2}-960\,{\pi }^{2}n_{{1}} \alpha _{{1}}n_{{2}}\alpha _{{2}}-480\,{\pi }^{2}{n_{{2}}}^{2}{\alpha _{{ 2}}}^{2})\mathfrak {I}_{1}(n)\end{aligned}$$

(47)

$$\begin{aligned} b_{1}=&\sum _{n=-\infty }^{\infty }(3840\,{\pi }^{4}{n_{{1}}}^{2}\alpha _{{1}}{n_{{2}}}^{2}{\alpha _{{2}}}^{ 2}\omega _{{1}}+3840\,{\pi }^{4}n_{{1}}\alpha _{{1}}{n_{{2}}}^{3}{\alpha _{{2}}}^{2}\omega _{{2}}-4096\,{\pi }^{6}{n_{{1}}}^{6}{\alpha _{{1}}}^{6 }\nonumber -24576\,{\pi }^{6}{n_{{1}}}^{5}{\alpha _{{1}}}^{5}n_{{2}}\alpha _{{2}}\\ {}&+\cdot \cdot \cdot \cdot +1280\,{\pi }^{4}{n_{{2}}} ^{4}{\alpha _{{2}}}^{3}\omega _{{2}}+80\,{\pi }^{2}{n_{{2}}}^{2}{\omega _ {{2}}}^{2} )\mathfrak {I}_{1}(n)\end{aligned}$$

(48)

$$\begin{aligned} a_{21}=&\sum _{n=-\infty }^{\infty }(16\,{\pi }^{2}\alpha _{{1}}n_{{1}}-16\,{\pi }^{2}\alpha _{{1}}{n_{{1}}}^ {2}-16\,{\pi }^{2}n_{{2}}\alpha _{{2}}n_{{1}}+8\,{\pi }^{2}n_{{2}} \alpha _{{2}}-4\,{\pi }^{2}\alpha _{{1}})\mathfrak {I}_{2}(n)\end{aligned}$$

(49)

$$\begin{aligned} a_{22}=&\sum _{n=-\infty }^{\infty }(8\,{\pi }^{2}\alpha _{{1}}n_{{2}}-16\,{\pi }^{2}\alpha _{{1}}n_{{1}}n_{{ 2}}-16\,{\pi }^{2}{n_{{2}}}^{2}\alpha _{{2}})\mathfrak {I}_{2}(n), a_{23}=\sum _{n=-\infty }^{\infty }\mathfrak {I}_{2}(n)\end{aligned}$$

(50)

$$\begin{aligned} a_{24}=&\sum _{n=-\infty }^{\infty }(-480\,{\pi }^{2}{\alpha _{{1}}}^{2}{n_{{1}}}^{2}+480\,{\pi }^{2}{\alpha _{{1}}}^{2}n_{{1}}-960\,{\pi }^{2}\alpha _{{1}}n_{{1}}n_{{2}}\alpha _{{2 }}\nonumber \\&-120\,{\pi }^{2}{\alpha _{{1}}}^{2}+480\,{\pi }^{2}\alpha _{{1}}n_{{2} }\alpha _{{2}}-480\,{\pi }^{2}{n_{{2}}}^{2}{\alpha _{{2}}}^{2})\mathfrak {I}_{2}(n)\end{aligned}$$

(51)

$$\begin{aligned} b_{2}=&\sum _{n=-\infty }^{\infty }(3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{3}n_{{2}}\alpha _{{2}} \omega _{{1}}-5760\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{2}n_{{2}} \alpha _{{2}}\omega _{{1}}+3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{ 2}{n_{{2}}}^{2}\alpha _{{2}}\omega _{{2}}\nonumber \\&+......+160\,{\pi }^{2}\omega _{{1}}n_{{1}}n_{{2}} \omega _{{2}}+1280\,{\pi }^{4}{\alpha _{{1}}}^{3}{n_{{1}}}^{3}n_{{2}}\omega _{{2}} )\mathfrak {I}_{2}(n)\end{aligned}$$

(52)

$$\begin{aligned} a_{31}=&\sum _{n=-\infty }^{\infty }(-16\,{\pi }^{2}n_{{2}}\alpha _{{2}}n_{{1}}-16\,{\pi }^{2}\alpha _{{1}}{n _{{1}}}^{2}+8\,{\pi }^{2}\alpha _{{2}}n_{{1}})\mathfrak {I}_{3}(n)\end{aligned}$$

(53)

$$\begin{aligned} a_{32}=&\sum _{n=-\infty }^{\infty }(8\,{\pi }^{2}\alpha _{{1}}n_{{1}}-16\,{\pi }^{2}{n_{{2}}}^{2}\alpha _{{2 }}+16\,{\pi }^{2}n_{{2}}\alpha _{{2}}-16\,{\pi }^{2}\alpha _{{1}}n_{{1}} n_{{2}}-4\,{\pi }^{2}\alpha _{{2}})\mathfrak {I}_{3}(n)\end{aligned}$$

(54)

$$\begin{aligned} a_{33}=&\sum _{n=-\infty }^{\infty }\mathfrak {I}_{3}(n)\end{aligned}$$

(55)

$$\begin{aligned} a_{34}=&\sum _{n=-\infty }^{\infty }(-480\,{\pi }^{2}{n_{{1}}}^{2}{\alpha _{{1}}}^{2}-960\,{\pi }^{2}n_{{1}} \alpha _{{1}}\alpha _{{2}}n_{{2}}+480\,{\pi }^{2}n_{{1}}\alpha _{{1}} \alpha _{{2}}-480\,{\pi }^{2}{\alpha _{{2}}}^{2}{n_{{2}}}^{2}\nonumber \\&+480\,{\pi }^{2}{\alpha _{{2}}}^{2}n_{{2}}-120\,{\pi }^{2}{\alpha _{{2}}}^{2})\mathfrak {I}_{3}(n) \end{aligned}$$

(56)

$$\begin{aligned} b_{3}=&\sum _{n=-\infty }^{\infty }(3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{3}n_{{2}}\alpha _{{2}} \omega _{{1}}+3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{2}{n_{{2}}}^ {2}\alpha _{{2}}\omega _{{2}}-3840\,{\pi }^{4}{n_{{1}}}^{2}{\alpha _{{1}} }^{2}\alpha _{{2}}n_{{2}}\omega _{{2}}\nonumber \\&+3840\,{\pi }^{4}\alpha _{{1}}{n_{{ 1}}}^{2}{n_{{2}}}^{2}{\alpha _{{2}}}^{2}\omega _{{1}}+.......+1280\,{\pi }^{4}{\alpha _{{1}}}^{3}{n_{{1}}}^{3}n_{{2}}\omega _{{2}})\mathfrak {I}_{3}(n) \end{aligned}$$

(57)

$$\begin{aligned} a_{41}=&\sum _{n=-\infty }^{\infty }(8\,{\pi }^{2}\alpha _{{2}}n_{{1}}+16\,{\pi }^{2}\alpha _{{1}}n_{{1}}-16 \,{\pi }^{2}\alpha _{{1}}{n_{{1}}}^{2}-16\,{\pi }^{2}\alpha _{{2}}n_{{2} }n_{{1}}-4\,{\pi }^{2}\alpha _{{2}}+8\,{\pi }^{2}\alpha _{{2}}n_{{2}}-4 \,{\pi }^{2}\alpha _{{1}} )\mathfrak {I}_{4}(n) \end{aligned}$$

(58)

$$\begin{aligned} a_{42}=&\sum _{n=-\infty }^{\infty }(16\,{\pi }^{2}\alpha _{{2}}n_{{2}}-16\,{\pi }^{2}\alpha _{{1}}n_{{1}}n_{ {2}}-16\,{\pi }^{2}\alpha _{{2}}{n_{{2}}}^{2}-4\,{\pi }^{2}\alpha _{{1}} -4\,{\pi }^{2}\alpha _{{2}}+8\,{\pi }^{2}\alpha _{{1}}n_{{2}}+8\,{\pi }^ {2}\alpha _{{1}}n_{{1}} )\mathfrak {I}_{4}(n)\end{aligned}$$

(59)

$$\begin{aligned} a_{43}=&\sum _{n=-\infty }^{\infty }\mathfrak {I}_{4}(n)\end{aligned}$$

(60)

$$\begin{aligned} a_{44}=&\sum _{n=-\infty }^{\infty }(480\,{\pi }^{2}{\alpha _{{2}}}^{2}n_{{2}}-240\,{\pi }^{2}\alpha _{{1}} \alpha _{{2}}-480\,{\pi }^{2}{\alpha _{{2}}}^{2}{n_{{2}}}^{2}-960\,{\pi }^{2}\alpha _{{1}}n_{{1}}\alpha _{{2}}n_{{2}}+480\,{\pi }^{2}{\alpha _{{1 }}}^{2}n_{{1}}\nonumber \\ {}&-120\,{\pi }^{2}{\alpha _{{1}}}^{2}-120\,{\pi }^{2}{ \alpha _{{2}}}^{2}+480\,{\pi }^{2}\alpha _{{1}}\alpha _{{2}}n_{{2}}+480\, {\pi }^{2}\alpha _{{1}}n_{{1}}\alpha _{{2}}-480\,{\pi }^{2}{\alpha _{{1}} }^{2}{n_{{1}}}^{2} )\mathfrak {I}_{4}(n)\end{aligned}$$

(61)

$$\begin{aligned} b_{4}=&\sum _{n=-\infty }^{\infty }(-5760\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{2}\alpha _{{2}}n_{{2}} \omega _{{1}}+3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}}}^{2}\alpha _{{2 }}{n_{{2}}}^{2}\omega _{{2}}\nonumber \\&-3840\,{\pi }^{4}{\alpha _{{1}}}^{2}{n_{{1}} }^{2}\alpha _{{2}}n_{{2}}\omega _{{2}}+\cdot \cdot \cdot \cdot -24576\,{\pi }^{6}{\alpha _{{1}}}^{ 5}{n_{{1}}}^{5}\alpha _{{2}}n_{{2}}+160\,{\pi }^{2}\omega _{{1}}n_{{1}} \omega _{{2}}n_{{2}} )\mathfrak {I}_{4}(n) \end{aligned}$$

(62)

where

$$\begin{aligned} \mathfrak {I}_{1}(n)&=\wp _{11}^{n_{1}^{2}}\wp _{12}^{2n_{1}n_{2}}\wp _{22}^{n_{2}^{2}},\nonumber \\ \mathfrak {I}_{2}(n)&=\wp _{11}^{n_{1}^2+(n_{1}-1)^{2}}\wp _{12}^{2[n_{1}n_2+(n_{1}-1)n_{2}]}\wp _{22}^{2n_{2}^{2}}\nonumber \\ \mathfrak {I}_{3}(n)&=\wp _{11}^{2n_{1}^{2}}\wp _{12}^{2[n_{1}n_{2}+n_{1}(n_{2}-1)]}\wp _{22}^{n_{2}^{2}+(n_{2}-1)^{2}},\nonumber \\ \mathfrak {I}_{4}(n)&=\wp _{11}^{n_{1}^{2}+(n_{1}-1)^2}\cdot \wp _{12}^{2[n_{1}n_{2}+(n_{1}-1)(n_{2}-1)]}\cdot \wp _{22}^{n_{2}^{2}+(n_{2}-1)^2}. \end{aligned}$$

(63)