Kadomtsev–Petviashvili hierarchy reduction, soliton and semi-rational solutions for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation in a fluid

Abstract

In this paper, we study a (3+1)-dimensional generalized shallow water wave equation with variable coefficients, which describes the flow below a pressure surface in a fluid. We give the Kadomtsev–Petviashvili hierarchy reduction and construct the multi-soliton solutions and semi-rational solutions in terms of the Gramian. For the multi-soliton solutions, we conclude that: (1) $v_1\left(t\right)$ affects the directions for the two solitons to move; (2) there is the periodic interaction of the two solitons when $v_2\left(t\right)$ is a periodic function and (3) the magnitudes of the velocities for the two solitons increase as the amplitude of the periodic function $v_2\left(t\right)$ increases, where $v_1\left(t\right)$ represents the perturbed effect, $v_2\left(t\right)$ indicates the dispersion effect and t is an independent variable. For the first-order semi-rational solutions, we see that: (1) the fission with $v_1\left(t\right)<0$ and fusion with $v_1\left(t\right)>0$ appear; (2) there is the periodic interaction when $v_2\left(t\right)$ is a periodic function; (3) the magnitude of the velocity for the soliton increases as the amplitude of the periodic function $v_2\left(t\right)$ increases and (4) the lump becomes narrower as the amplitude of the periodic function $v_4\left(t\right)$ decreases, where $v_4\left(t\right)$ indicates the perturbed effect.

Keywords:

• Fluid
• (3+1)-dimensional generalized shallow water wave equation
• semi-rational solutions
• soliton solutions
• Kadomtsev–Petviashvili hierarchy reduction

2010 Mathematics Subject Classifications:

• 35C06
• 35C08

Acknowledgments

We express our sincere thanks to the Editors and Reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A polynomial ${\tau }_n\left(\varsigma \right)$ is a tau function if and only if it satisfies $\oint e^{\eta (\varsigma -{\varsigma }^{\prime },b)}{\tau }_n[\varsigma -\mathrm{\Theta }(b\left)\right]{\tau }_n[{\varsigma }^{\prime }+\mathrm{\Theta }(b\left)\right]\frac{db}{2\pi i}=0$ (for any ς, ${\varsigma }^{\prime }$), and the integration is taken along a small contour at $b=\mathrm{\infty }$ so that $\oint \frac{db}{2\pi ib}=1$, where $i^2=-1$, ϖ is a positive integer, $x_{\varpi }$'s, $x_{\varpi }^{\prime }$'s and b are the independent variables, $\varsigma =(x_1,x_2,\dots ,x_{\varpi },\dots )$, ${\varsigma }^{\prime }=(x_1^{\prime },x_2^{\prime },\dots ,x_{\varpi }^{\prime },\dots )$ and $\mathrm{\Theta }\left(b\right)=(\frac{1}{b},\frac{1}{2b^2},\dots ,\frac{1}{\varpi b^{\varpi }},\dots )$ are the vectors, $\eta (\varsigma ,b)=\sum _{\varpi =1}^{\mathrm{\infty }}{b}^{\varpi }{x}_{\varpi }$ [25].

2 (Bilinear) KP hierarchy is defined as a series of the bilinear forms satisfying $\left[\sum _{h=0}^{\mathrm{\infty }}{\chi }_h(-2\text{β}){\chi }_{h+1}\right(\tilde{D}\left)exp\right(\sum _{\kappa =1}^{\mathrm{\infty }}{\text{β}}_{\kappa }{D}_{\kappa }\left)\right]{\tau }_n\cdot {\tau }_n=0$ (for any β), where h is a non-negative integer, κ is a positive integer, ${\text{β}}_{\kappa }$'s is the independent variable, ${\chi }_h$'s is the coefficient polynomial, $\text{β}=({\text{β}}_1,{\text{β}}_2,\dots ,{\text{β}}_{\kappa },\dots )$ is a vector, $\tilde{D}=(D_1,\frac{D_2}{2},\dots ,\frac{D_{\kappa }}{\kappa }\cdots )$, $D_{\kappa }$'s is the bilinear derivative operator [25].

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11272023,11772017,11805020] and Fundamental Research Funds for the Central Universities of China [2011BUPTYB02] and State Key Laboratory of Information Photonics and Optical Communications [2017ZZ05] and Fundamental Research Funds for the Central Universities in UIBE (CXTD12-04).