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

Abstract

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.

Quasi-integrable KdV models, towers of infinite number of anomalous charges and soliton collisions

“…We will find below other type of anomalous charges for the standard KdV. These new kind of anomalous charges are expected to appear in the other quasi-integrable theories considered in the literature [11].…”

“…These new charges differ in form from the relevant charges corresponding to the undeformed model. As it has been mentioned in [11], an infinite subset of those new charges turned out to be anomalous even for the standard sine-Gordon model.…”

“…'s of the relevant quasi-conservation laws (3.1) can be expressed as time-derivatives of the relevant KdV-type charges. Fourth, the behavior above is in contradistinction to the deformed sine-Gordon models, in which the analogous higher order quasi-conservation laws, in the anomalous zero-curvature approach, become simply the higher derivatives of the non-trivial energy-momentum conservation law [11].…”

“…Several new towers of infinite number of anomalous conservation laws for deformed sine-Gordon models have been uncovered by direct construction [11]. Remarkably, it has been observed that even the standard sine-Gordon model possesses those types of anomalous charges for soliton configurations satisfying the special space-time inversion symmetry properties.…”

“…The models considered in [11] were deformed sine-Gordon models; i.e. relativistic models with topological solitons.…”