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 {ϵ1, ϵ2}, which include the standard KdV (ϵ1 = ϵ2 = 0), the regularized long-wave (RLW) (ϵ1 = 1, ϵ2 = 0), the modified regularized long-wave (mRLW) (ϵ1 = ϵ2 = 1) and the KdV-RLW (KdV-BBM) type (ϵ2 = 0, ≠ = {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 {ϵ1, ϵ2}, 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.
Highlights
Quasi-integrability concept related to the anomalous zero-curvature approach to modifications of integrable models [1,2,3]
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 exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for N −soliton solution
Summary
We will consider the model studied in [10] as a particular deformation of the KdV equation. A direct method provides a general 1-soliton solution of (2.6) by assuming the form ζ qII = q0 log cosh 2a + bζ + c , ζ = kx − w2t + δ Such that a, b and c are arbitrary real parameters. In order to implement the parity transformation (2.11) and check the space-time parity inversion symmetry of the 2-soliton solution we will derive a new expression for q in (2.22), Such that u2-sol in (2.4) becomes a manifestly P invariant function. In [10] it was provided a different procedure to construct u2-sol, and it has been shown that the exact Hirota three-soliton solutions of the standard KdV equation possess the relevant parity properties when their solitons collide at the same point in space
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