Solutions of the Bullough–Dodd Model of Scalar Field through Jacobi-Type Equations

Abstract

A technique based on multiple auxiliary equations is used to investigate the traveling wave solutions of the Bullough–Dodd (BD) model of the scalar field. We place the model in a flat and homogeneous space, considering a symmetry reduction to a 2D-nonlinear equation. It is solved through this refined version of the auxiliary equation technique, and multiparametric solutions are found. The key idea is that the general elliptic equation, considered here as an auxiliary equation, degenerates under some special conditions into subequations involving fewer parameters. Using these subequations, we successfully construct, in a unitary way, a series of solutions for the BD equation, part of them not yet reported. The technique of multiple auxiliary equations could be employed to handle several other types of nonlinear equations, from QFT and from various other scientific areas.

The auxiliary equation approach for solving reaction-diffusion equations

“…Our study was devoted to the most general form of a reaction-diffusion equation whose solutions can be expressed in terms of Riccati's solutions. We started from a reaction-diffusion equation of the form (2) with the first three polynomial coefficients of the form (6). The compatibility between a general reaction-diffusion equation and the Riccati equation relies on the particular circumstances and connections between the coefficients and parameters in the equations.…”

“…Depending on how the previous steps are applied, the solving methods can be classified accordingly. For example, following the choice of the auxiliary equation, the most used equations chosen to fulfill this role are the Riccati equation and the elliptic Jacobi equation [6], but many other equations have been also used. Following the form in which the solutions are sought, different approaches were proposed and intensively used in literature: sine-cosine method [7], tanh-method [8], Kudryashov method [9,10], the functional expansion method [11], exp-function method [12], (G ′ /G)-method [13], the attached flow approach [14], etc.…”

Attached Flows for Reaction–Diffusion Processes Described by a Generalized Dodd–Bullough–Mikhailov Equation

“…At the classical level, ( 12) is known as a nonlinear integrable model describing both classical and quantum propagation phenomena. In the quantum context, the equation appears as the Bullough-Dodd model for scalar fields [15], and it is equivalent to an affine Toda field with zero curvature representation in twisted affine Kac-Moody algebras [16]. This feature assures the existence of soliton-like solutions.…”

“…The simplest approaches tend to use a predefined form of solutions, as in [19] or [20], where the tanh method is applied. More generally, traveling wave solutions for DBM have been generated via different methods belonging to the auxiliary equation technique: the G ′ G -expansion [21], the functional expansion [22], or the sub-equation technique, considering the elliptic Jacobi as an auxiliary equation [15]. Soliton and compacton-like solutions were generated in [23] with the exponential function method, or in [24]with the generalized Kudryashov method and an improved F-expansion method.…”