Pseudo-potenciais de tipo Riccati, leis de quase-conservação e sólitons do modelo deformado de seno-Gordon

Abstract

Deformed sine-Gordon (DSG) models ∂ξ∂ηw+ 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. In addition, we provide a linear formulation 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-Zakrzewski, JHEP05(2011)130], possess new towers of infinite number of quasi-conservation laws. These new quasi-conservation laws are also obtained in a rigorous way in the context of Riccati type pseudopotential formalism. 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 et al. potential Vq(w) = 64 q 2 tan2 w 2 (1 − |sin w 2 | 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.