Construction of abundant solutions for two kinds of $\mathbf {(3\varvec{+}1)}$-dimensional equations with time-dependent coefficients

Abstract

Variable-coefficient nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and non-uniformities of boundaries than their constant coefficients in some real-world problems. A $(3+1)$-dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa (vcDJKM) equation and a $(3+1)$-dimensional variable coefficient Boiti–Leon–Manna–Pempinelli (vcBLMP) equation are studied based on the homoclinic test method and construction of abundant solutions for two kinds of equations are obtained with the help of symbolic computation. In order to study the interaction superposition solutions with different functions, an existence theorem and corollary about superposition solutions of the $(3+1)$-dimensional vcBLMP equation are proved. Via three-dimensional profiles with the help of mathematics, the propagation and the dynamical behavior of these solutions are analyzed by choosing different arbitrary variable coefficients. Compare with the published studied, some completely new results are presented in this paper.

Appendix

$$\begin{aligned} \begin{aligned} A_{1}(t)&= (a_{1}^{2}p_{1}^{2}+a_{2}^{2}p_{2}^{2} )^{2}[h(t)(a_{1}\lambda _{1}+b_{1}\lambda _{2}+c_{1}\lambda _{3})\\&\quad +a_{1}^{2}b_{1}p_{1}^{2} -a_{2}p_{2}^{2}(a_{2}b_{1}+2a_{1}b_{2})] \\&\quad +\alpha [a_{2}b_{2}^{2}p_{2}^{4}(2a_{1}b_{2}-3a_{2}b_{1}) +b_{1}p_{1}^{2}p_{2}^{2}(a_{2}^{2}b_{1}^{2}\\&\quad +3a_{1}^{2}b_{2}^{2} -6a_{1}a_{2}b_{1}b_{2})-a_{1}^{2}b_{1}^{3}p_{1}^{4}], \\ \end{aligned} \end{aligned}$$

$$\begin{aligned} A_{2}(t)&= (a_{1}^{2}p_{1}^{2}+a_{2}^{2}p_{2}^{2} )^{2}[h(t)(a_{2}\lambda _{1}+b_{2}\lambda _{2}\nonumber \\&\quad +c_{2}\lambda _{3})-a_{2}^{2}b_{2}p_{2}^{2} +a_{1}p_{1}^{2}(2a_{2}b_{1}+a_{1}b_{2})] \nonumber \\&\quad +\alpha [a_{1}b_{1}^{2}p_{1}^{4}(2a_{2}b_{1}-3a_{1}b_{2})\nonumber \\&\quad +b_{2}p_{1}^{2}p_{2}^{2}(3a_{2}^{2}b_{1}^{2}\nonumber \\&\quad +a_{1}^{2}b_{2}^{2} -6a_{1}a_{2}b_{1}b_{2})-a_{2}^{2}b_{2}^{3}p_{2}^{4}], \nonumber \\ A_{3}(t)&= \alpha [2a_{1}a_{2}b_{2}(b_{2}^{2}-3b_{1}^{2})+b_{1}(a_{1}^{2}-a_{2}^{2})(3b_{2}^{2}-b_{1}^{2})] \nonumber \\&\quad +h(t)(a_{1}^{2}+a_{2}^{2})^{2}(a_{1}\lambda _{1}+b_{1}\lambda _{2}+c_{1}\lambda _{3})\nonumber \\&\quad +(a_{1}^{2}+a_{2}^{2})^{2}p^{2}(a_{1}^{2}b_{1}-a_{2}^{2}b_{1}-2a_{1}a_{2}b_{2}), \nonumber \\ A_{4}(t)&= \alpha [2a_{1}a_{2}b_{1}(b_{1}^{2}-3b_{2}^{2})+b_{2}(a_{1}^{2}-a_{2}^{2})(b_{2}^{2}-3b_{1}^{2})] \nonumber \\&\quad +h(t)(a_{1}^{2}+a_{2}^{2})^{2}(a_{2}\lambda _{1}+b_{2}\lambda _{2}+c_{2}\lambda _{3})\nonumber \\&\quad +(a_{1}^{2}+a_{2}^{2})^{2}p^{2}(2a_{1}a_{2}b_{1}+a_{1}^{2}b_{2}-a_{2}^{2}b_{2}), \nonumber \\ A_{5}(t)&= \alpha [2a_{1}a_{2}b_{2}(b_{2}^{2}-3b_{1}^{2})+b_{1}(a_{1}^{2}-a_{2}^{2})(3b_{2}^{2}-b_{1}^{2})]\nonumber \\&\quad +h(t)(a_{1}^{2}+a_{2}^{2})^{2} (a_{1}\lambda _{1}+b_{1}\lambda _{2}+c_{1}\lambda _{3}), \nonumber \\ A_{6}(t)&= \alpha [2a_{1}a_{2}b_{1}(b_{1}^{2}-3b_{2}^{2})+b_{2}(a_{1}^{2}-a_{2}^{2})(b_{2}^{2}-3b_{1}^{2})]\nonumber \\&\quad +h(t)(a_{1}^{2}+a_{2}^{2})^{2} (a_{2}\lambda _{1}+b_{2}\lambda _{2}+c_{2}\lambda _{3}), \nonumber \\ A_{7}(t)&= (a_{3}^{2}p_{3}^{2}-a_{4}^{2}p_{4}^{2} )^{2}[h(t)(a_{3}\lambda _{1}+b_{3}\lambda _{2}\nonumber \\&\quad +c_{3}\lambda _{3})+a_{3}^{2}b_{3}p_{3}^{2} +a_{4}p_{4}^{2}(a_{4}b_{3}+2a_{3}b_{4})] \nonumber \\&\quad -\alpha [a_{4}b_{4}^{2}p_{4}^{4}(3a_{4}b_{3}-2a_{3}b_{4})+b_{3}p_{3}^{2}p_{4}^{2}(a_{4}^{2}b_{3}^{2}\nonumber \\&\quad +3a_{3}^{2}b_{4}^{2} -6a_{3}a_{4}b_{3}b_{4})+a_{3}^{2}b_{3}^{3}p_{3}^{4}], \nonumber \\ A_{8}(t)&= (a_{3}^{2}p_{3}^{2}-a_{4}^{2}p_{4}^{2} )^{2}[h(t)(a_{4}\lambda _{1}+b_{4}\lambda _{2}\nonumber \\&\quad +c_{4}\lambda _{3})+a_{4}^{2}b_{4}p_{4}^{2} +a_{3}p_{3}^{2}(2a_{4}b_{3}+a_{3}b_{4})] \nonumber \\&\quad -\alpha [a_{3}b_{3}^{2}p_{3}^{4}(3a_{3}b_{4}-2a_{4}b_{3})+b_{4}p_{3}^{2}p_{4}^{2}(a_{3}^{2}b_{4}^{2}\nonumber \\&\quad +3a_{4}^{2}b_{3}^{2}-6a_{3}a_{4}b_{3}b_{4})+a_{4}^{2}b_{4}^{3}p_{4}^{4}], \nonumber \\ A_{9}(t)&=(p_{1}^{2}+p_{2}^{2} )^{2}[h(t)b_{1}+b_{1}p_{1}^{2}-p_{2}^{2}(b_{1}+2b_{2})] \nonumber \\&\quad +\alpha [b_{2}^{2}p_{2}^{4}(2b_{2}-3b_{1})+b_{1}p_{1}^{2}p_{2}^{2}(b_{1}^{2}\nonumber \\&\quad +3b_{2}^{2}-6b_{1}b_{2})-b_{1}^{3}p_{1}^{4}], \nonumber \\ A_{10}(t)&= (p_{1}^{2}+p_{2}^{2} )^{2}[h(t)b_{2}-b_{2}p_{2}^{2}+p_{1}^{2}(2b_{1}+b_{2})] \nonumber \\&\quad +\alpha [b_{1}^{2}p_{1}^{4}(2b_{1}-3b_{2})+b_{2}p_{1}^{2}p_{2}^{2}(3b_{1}^{2}\nonumber \\&\quad +b_{2}^{2}-6b_{1}b_{2})-b_{2}^{3}p_{2}^{4}]. \end{aligned}$$

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