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.
Highlights
IntroductionA model of a scalar field, u( x μ ), in Quantum Field Theory (QFT) can be described through a Lagrangian density of the form: Jacobi-Type Equations
Model of Scalar Field throughA model of a scalar field, u( x μ ), in Quantum Field Theory (QFT) can be described through a Lagrangian density of the form: Jacobi-Type Equations
By solving the appropriate algebraic system by making use of the Maple program, the links that exist between the previous parameters and the ones that are related to the dynamical model (14) can be derived; Step 4: By taking into consideration the results that we obtained through the above steps, a series of traveling wave solutions of Equation (16) that depend on the solutions of the generalized auxiliary Equation (5) can be derived
Summary
A model of a scalar field, u( x μ ), in Quantum Field Theory (QFT) can be described through a Lagrangian density of the form: Jacobi-Type Equations. The focus of this paper is on how such traveling waves can be obtained using a very general approach, based on the auxiliary equation technique It consists of looking for solutions of a complicated nonlinear partial differential equation (NPDE) in terms of the known solutions of an “auxiliary equation”. Despite the numerous studies on the BD equation and the various techniques used to solve it, it seems that the model contains a much richer phenomenology, and as we will see, our investigation allows generating new classes of solutions, depending on a larger number of parameters. Various solutions of the algebraic system generated among all these parameters give us new, more complex, BD solutions, and their dependence on the specific forms of auxiliary equations As we mentioned, these equations represent reductions of the general elliptic equation with five parameters. Some essential facts are pointed out as concluding remarks
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