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

Abstract

Deformed sine-Gordon (DSG) models ∂ξ∂ηw+ddwV(w)=0, with V(w) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential 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 the DSG model and an associated 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 and Zakrzewski, 2011 [1]), 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 pseudo-potential 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 et al. potential Vq(w)=64q2tan2⁡w2(1−|sin⁡w2|q)2(q∈R), which contains the usual SG potential V2(w)=2[1−cos⁡(2w)]. The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.