Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients

Abstract

In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined by the linear part of the equation. These conditions generalize the ones given by the author for the linear case [?,?] and for the semilinear heat equation [?]. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.Summability, Inhomogeneous partial differential equation, Nonlinear partial differential equation, Formal power series, Divergent power series 35C10, 35C20, 40B05

Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

“…For example, in the case of real variables, that is when the variables pt, xq belong to a subset of R 2 , we can quote, among the many existing techniques, the tanh-sech method, the F-expansion method, the exp-function method, the variational iteration method, etc. More recently, in the case of complex variables, that is when the variables pt, xq belong to a subset of C 2 , the summation theory has also been used succesfullly [19,25,26]. This theory, initially developed within the framework of the meromorphic ordinary differential equation with an irregular singular point (see for instance [10,16]), allows the construction of explicit solutions from formal solutions.…”

“…Known results and aim of the article. In the two previous articles [18,26] (see also [20], and [17,19,24] for more general equations), the author studied, in the framework of the Euler-Lagrange equation (1.3), the Gevrey regularity and the 1-summability of the formal solution r upt, xq. More precisely, he proved the two following.…”

“…(1.3). Then, Proposition 1.3 (Summability [26]). Let r upt, xq be the formal solution in OpD ρ1 qrrtss of Eq.…”

“…In this article, we propose to prove that these two results remain true in the case of the generalized Boussinesq equation (1.2). To do this, we shall use an approach similar to those already developed by the author in [17][18][19][23][24][25][26] for some nonlinear partial differential equations (see also [3,[20][21][22] for an approach in the linear case). Let us point out here that, as we shall see below, the terms u m pB x uq 2 of Eq.…”