Paper detail

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

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 $\{ε_1, ε_2\}$, which include the standard KdV ($ε_1=ε_2=0$), the regularized long-wave (RLW) ($ε_1=1,ε_2=0$), the modified regularized long-wave (mRLW) ($ε_1=ε_2=1$) and the KdV-RLW (KdV-BBM) type ($ε_2=0,\,ε\neq \{0,1\}$) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set $\{ε_1, ε_2\}$, 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 $AdS_{3}$, Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.