Disclosing the generic behavior of topological solutions: An orbit-based approach

Abstract

In this work we present a method, based on the vacuum structure of the potential for a system of two nonlinearly coupled scalar fields in 1 + 1 space-time dimensions, which yields complete information about the behavior of the topological configurations. This is done by means of an analysis of the orbits and the position of the degenerate vacua of the model on the configuration space.

Wide vector solitons in systems with time- and space-modulated nonlinearities

“…Finally, this analogy allows us to use the orbit equation method [19] to find topological solutions of the two coupled autonomous CQNLSEs. With this method an integration constant shows up naturally, which controls the width and the amplitude of the topological solution when the constant is close to its critical value [20]. As we show, this strategy leads to several solutions which can be continuously deformed into wide vector solitons by tuning the integration constant.…”

“…We refer the reader to Refs. [19], [20], and [26] to appreciate the details of the role played by c 0 in the variety of solutions and the consequences of such solutions in specific models.…”

Orbit based procedure for doublets of scalar fields and the emergence of triple kinks and other defects

“…An interesting fact is that it is possible to obtain analytical solutions for this model, unlike other models with sixth degree terms on its potential [31]. Note that it is possible to use the above orbit in order to express φ in terms of χ and then substitute it in (31). Performing some changes of variables we may integrate the remaining first order equation to obtain…”

“…thus, the corresponding coordinates of the vacua states in the internal space may be written as follows It is not difficult to verify that the vacua states v 1 and v 2 satisfy (39) independently of the value chosen for the constant c. Otherwise, other vacua states satisfies the orbit (39) only for some critical value of c that is determined by the substitution of coordinates of the vacua sates in the orbit solution [31]. For instance, let us consider the vacua v 3 , the critical value c 0 is given by…”

“…Recently, it was shown that one can go further by performing a deformation of the orbit equation [30]. On the other hand, at least partially, one can devise the general behavior of the topological solutions of a given nonlinear model by studying its vacuum structure and its orbits [31]. In the last reference, it was noticed that the appearance of double-kink and flat-top lumps, was a consequence of the passage of the orbit in the vicinity of a vacuum, before to go to another one.…”