Convergence for Jacobi elliptic function series solutions to one kind of perturbed Kadomtsev-Petviashvili equations

Abstract

This paper is devoted to constructing series solutions to one kind of perturbed Kadomtsev-Petviashvili (KP) equations, of which the perturbation terms are of all six-order derivatives of space variable <inline-formula><tex-math id="M10">\begin{document}$x$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M10.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$y$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M11.png"/></alternatives></inline-formula>. First, by making the series solutions expansion with respect to the homotopy parameter <inline-formula><tex-math id="M12">\begin{document}$q$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M12.png"/></alternatives></inline-formula>, the homotopy model of the perturbed KP equations can be decomposed into infinite number of approximate equations of the general form. Second, Lie symmetry method is applied to these approximate equations to achieve similarity solutions and the related similarity equations with common formulae in three cases. Third, for the first few similarity equations in the third case, Jacobi elliptic function solutions are constructed through a step-by-step procedure and are also subject to common formulae for each equation of the whole kind of perturbed KP equations. Finally, one kind of compact series solutions for the original perturbed KP equations is obtained from these Jacobi elliptic function solutions. The convergence of these series solution is dependent on perturbation parameter <inline-formula><tex-math id="M13">\begin{document}$\epsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M13.png"/></alternatives></inline-formula>, auxiliary parameter <inline-formula><tex-math id="M14">\begin{document}$\theta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M14.png"/></alternatives></inline-formula> and arbitrary constants <inline-formula><tex-math id="M15">\begin{document}$\{a, b, c\}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M15.png"/></alternatives></inline-formula>, among which the most prominent is decreasing arbitrary constant <inline-formula><tex-math id="M16">\begin{document}$c$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M16.png"/></alternatives></inline-formula> or perturbation parameter <inline-formula><tex-math id="M17">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M17.png"/></alternatives></inline-formula>. For the perturbation term in perturbed KP equations, given the derivative order <inline-formula><tex-math id="M18">\begin{document}$n$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M18.png"/></alternatives></inline-formula> of <inline-formula><tex-math id="M19">\begin{document}$u$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M19.png"/></alternatives></inline-formula> with respect to <inline-formula><tex-math id="M20">\begin{document}$y$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M20.png"/></alternatives></inline-formula>, smaller (greater) <inline-formula><tex-math id="M21">\begin{document}$|a/b|$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M21.png"/></alternatives></inline-formula> causes the improved convergence provided <inline-formula><tex-math id="M22">\begin{document}$n\leqslant 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M22.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M23">\begin{document}$n\geqslant 3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M23.png"/></alternatives></inline-formula>). Nonetheless, the decrease of arbitrary constant <inline-formula><tex-math id="M24">\begin{document}$|c|$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M24.png"/></alternatives></inline-formula> or <inline-formula><tex-math id="M25">\begin{document}$|a/b|$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20190333_M25.png"/></alternatives></inline-formula> leads to the enlargement of period in a certain direction and thus should be specified appropriately. This paper also considers the perturbed KP equations with more general perturbation terms. Only if the derivative order of the perturbation term is an even number, do Jacobi elliptic function series solutions exist for perturbed KP equations. The existence of series solutions can serve as a criterion of solvability for perturbed equations.