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.

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Acknowledgements

The authors deeply appreciate the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further. This work is supported by the National Natural Science Foundation of China (Grant No. 11361040), the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2020LH01008), the Graduate Students’ Scientific Research Innovation Fund Program of Inner Mongolia Normal University, China (Grant Nos. CXJJS19096, CXJJS20089) and the Graduate Research Innovation Project of Inner Mongolia Autonomous Region, China (Grant No. S20191235Z).

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Appendix

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}$$
(45)

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Han, PF., Bao, T. Construction of abundant solutions for two kinds of \(\mathbf {(3\varvec{+}1)}\)-dimensional equations with time-dependent coefficients. Nonlinear Dyn (2021). https://doi.org/10.1007/s11071-020-06167-4

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Keywords

  • \((3+1)\)-Dimensional vcDJKM equation
  • \((3+1)\)-Dimensional vcBLMP equation
  • Hirota bilinear method
  • Homoclinic test method
  • Breather-kink wave
  • Rogue wave
  • Three-solitary wave
  • Compound solutions of different functions