Some deformations of the integrable Korteweg-de Vries model (KdV) are associated to several towers of infinite number of asymptotically conserved charges. It has been shown that the standard KdV also exhibits infinite number of anomalous charges. In [9] there have been verified numerically the degrees of modifications of the charges around the soliton interaction regions, by computing numerically some representative anomalies, related to lowest order quasi-conservation laws, depending on the deformation parameters { ∈ 1 , ∈ 2 } , such that the standard KdV is recovered for ( ∈ 1 = ∈ 2 = 0 ) . Here we present the numerical simulations for some values of the pair { ∈ 1 , ∈ 2 } around ∈ 1 ≈ 0 , ∈ 2 ≈ 0 , and show that the collision of two and three solitons are elastic. The KdV-type equations are quite ubiquitous and find many applications in several areas of non-linear science.

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.

The objective of this paper was to find the interaction between two vortex solitons in the Nielsen – Olensen model, these solitons are the asymptotic solutions in said model, in order to calculate this interaction it was assumed that these solitons are quasi-static and that they are very far between if and with the help of the momentum energy tensor, this interaction could be found in the z direction in a plane that symmetrically divides these two solitons, which presents two cases since there are two possible asymptotic solutions for the field function according to the coefficient β, the possible cases are for β < 4 and β > 4, in each of the cases interaction is found that depends on exponential functions, the distance between these solitons and the parameters found at the time of calculating said asymptotic solution that are constants {1, −1}. Furthermore, this force is similar to the force experienced by two turns with current that are very far from each other.