Deformed sine-Gordon (DSG) models ∂ ξ ∂η w + d dw V (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-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 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) = 64 q 2 tan 2 w 2 (1 − | sin w 2 | q ) 2 (q ∈ IR), 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.
We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential V q w = 64 q 2 tan 2 w 2 1 − sin w 2 q 2 , q ∈ R , which contains the usual SG potential V 2 w = 2 1 − cos 2 w .
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