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

Abstract

We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for N −soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the degrees of modifications of the charges around the soliton interaction regions, we compute numerically some representative anomalies, associated to the lowest order quasi-conservation laws, depending on the deformation parameters {ǫ1, ǫ2}, which include the standard KdV (ǫ1 = ǫ2 = 0), the regularized long-wave (RLW) (ǫ1 = 1, ǫ2 = 0), the modified regularized long-wave (mRLW) (ǫ1 = ǫ2 = 1) and the KdV-RLW (KdV-BBM) type (ǫ2 = 0, ǫ = {0, 1}) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set {ǫ1, ǫ2}, for a variety of amplitudes and relative velocities. The KdV-type equations are quite ubiquitous in several areas of non-linear science, and they find relevant applications in the study of General Relativity on AdS3, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

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

“…1) the anomalies a + and γ − vanish. These kind of anomalous charges also appear in the standard sine-Gordon model, and they are expected to appear in the other integrable systems and their quasi-integrable deformations [11,12].…”

“…The only analytical explanation we have found, so far, for the unexpected appearance of these anomalous charges are the space-time symmetry properties which the 2-soliton solutions of the standard sine-Gordon model exhibit. It is expected that those types of charges will play an important role in the study of soliton gases and formation of certain structures in (quasi-)integrable systems, such as soliton turbulence, soliton gas dynamics and rogue waves [11,12]. In addition, these new kind of charges are expected to appear in the other quasi-integrable theories considered in the literature.…”

Deformed Sine-Gordon Models, Solitons and Anomalous Charges

“…In this context the so-called quasi-integrability concept has been put forward [4]. These properties have been examined in the frameworks of the anomalous zero-curvature [4][5][6][7] and the Riccati-type pseudo-potential approaches [8][9][10], respectively.…”

“…Fifth, there exist infinite towers of infinitely many anomalous charges, different in form from the ones of the usual integrable models. New towers of anomalous charges have been uncovered in [8][9][10]. Remarkably, even the usual integrable models possess quasi-conservation laws with anomalous charges for analytical NÀ soliton with CP s T d symmetry [9,10].…”

“…New towers of anomalous charges have been uncovered in [8][9][10]. Remarkably, even the usual integrable models possess quasi-conservation laws with anomalous charges for analytical NÀ soliton with CP s T d symmetry [9,10]. For the standard SG theory it has been discussed for the 2-soliton sector of the theory [8].…”