A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

Abstract

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a Bäcklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

“…After this, some traveling wave solutions, both regular and singular, have been rederived by Darboux or other direct methods (for this active line of research see, for example, [17][18][19][20]). However, these kind of studies do not give any insight as regards the solution to a generic IVP with decaying Cauchy data, or provide an useful tool to classify the manifold of all real, regular, and bounded solutions.…”

Преобразования Двойственности Для Спектральной Задачи В Размерности $2+1$

“…Тем не менее иногда можно указать нетривиальные преобразования, которые приводят уравнение к виду, допускающему применение метода Пенлеве. Например, в работе [5] мы применили преобразование двойственности [6]- [9] к (2 + 1)-мерной иерархии Камассы-Холма, что позволило преобразовать ее к системе уравнений, для которых можно с успехом применить метод сингулярного многообразия [10].…”

Singular Manifold Method for an Equation in 2 + 1 Dimensions

“…Nevertheless, as it has been proved in [7], the SMM can be implemented by including both branches simultaneously. This generalization of the SMM has been applied successfully to many equations with two Painlevé branches [9], [10].…”

“…Modifications of SMM that includes the different branches simultaneously can be found in the following references: [7], [9], [10] and [12]. Once more the solution of this problem includes a bonus: If an equation has two Painlevé branches, the modification of the SMM provides us not only the right answer but the Miura transformations that relate our initial PDE to another PDE with just one Painlevé branch ( [9], [10]). …”

“…Actually, when we have been able to write a PDE in a form in which the SMM has been successful, this method provides us the singular manifold equations that can be considered as the canonical form of a PDE. We can conjecture that, if two PDE's have the same singular manifold equations, then there exists a transformation that relates the two equations since they are essentially the same [10].…”