Riccati-type pseudopotentials, conservation laws and solitons of deformed sine-Gordon models

Abstract

Deformed sine-Gordon (DSG) models $\partial_\xi \partial_\eta \, w + \frac{d}{dw}V(w) = 0$, with $V(w)$ being the deformed potential, are considered in the context of the Riccati-type pseudopotential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for DSG model, and provide an infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach [Ferreira-Zakrzewski, JHEP05(2011)130], possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in the pseudopotential approach, and the first four anomalies of the new towers of charges, respectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia {\sl et al.} potential $V_{q}(w) = \frac{64}{q^2} \tan^2{\frac{w}{2}} (1-|\sin{\frac{w}{2}}|^q)^2 \, (q \in R)$, which contains the usual SG potential $V_2(w) = 2[1- \cos{(2 w)}]$. The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.