Mapping for BPS Solitons of Scalar Field Potentials in 1 + 1 Dimensions and Family of Solutions

We analyze a method for solving a second-order nonlinear differential equation in 1 + 1 dimensions, applying it to some nonlinear systems. A particular solution for systems with this dimensionality is known as kink. In this study, we focus on revealing that any kink in 1 + 1 dimensions, accruing from models with one scalar field, can be straightforwardly obtained from a scalar field solution to a first-order linear differential equation with constant coefficients. This is accomplished by a suitable field transformation and we examine a few models and analyze how the introduction of an underlying scalar field can shed new light on models with one scalar field. In this work, in contrast to what is expected, we show that any kink in (1 + 1) dimensions, originating from models with just one scalar field, can be obtained from a master linear first-order differential equation using a convenient field transformation, which leads to a linear differential equation for the transformation function. A general approach is introduced and discussed, including a few subsequent cogent and important physical applications. This approach for certain values of parameters presents symmetry breaking like the λϕ4 model. The other parameter values correspond to a model with no minima, presenting kink configurations for the scalar field. In this study, we focus on revealing that any kink in (1 + 1) dimensions, accruing from models with one scalar field, can be obtained from the master linear first-order differential equation. This is accomplished through a convenient field transformation, obeying a linear differential equation for the transformation function. After analyzing a few models, we present a new one using the method developed in this work.