A Method for Solving Nonlinear Differential Equations: An Application to λφ4 Model

Abstract

Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Padé approximants for solving nonlinear partial differential equations without requiring a one-dimensional reduction. This method is applied to the λφ 4 model in 4 dimensions and new solutions are obtained.

Classical solution for the linear sigma model

“…With the solutions found for two and three bosonic fields, we were able to write a solution for an arbitrary number of bosons and traveling waves. The results of the current work show that the method developed in [20] is robust and can be used to find explicit solutions in specific problems in classical field theory.…”

“…In these system, we impose c 00 = d 00 = 0 because we would like that the fields be null at infinite (similarly, this physical constraint could be imposed on ansats (20) for simplify the solution of the algebraic system). So, the above system yields c 00 = d 00 = 0, k 1µ k µ 1 +m 2 = 0, k 2µ k µ 2 +m 2 = 0, c 10 , c 01 , d 10 , d 01 arbitrary.…”

“…The purpose of this paper is to seek classical solutions for this system in the form of traveling waves. In order to achieve that, we will employ the algorithm based on Padé approximants presented in [19,20]. This method has the advantage of needing less undetermined constants than other methods in the literature [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], such that we can earn efficiency in the processing of data.…”

“…This method has the advantage of needing less undetermined constants than other methods in the literature [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], such that we can earn efficiency in the processing of data. We briefly recall the algorithm from [20] in section (2) and we apply it to the linear sigma model in section (3). Using the solutions for two and three traveling waves we found, we are able to show a solution for an arbitrary number of traveling waves and bosons in section (3.3).…”