Derivation of an Equation for the Correction to the Soliton KdV Velocity by the Weak Asymptotic Method

The equations describing soliton dynamics in the framework of the Maslov – Whitham method have long been well known. However, the Maslow –Whitham method itself is not applicable for describing the interaction of solitons , especially in non-integrable versions of KdV . The problem of describing the interaction of solitons in the framework of the weak asymptotic method was solved in the works of V.G. Danilova and G.A.Omelyanova. In these papers, conditions were found under which the scenario of the interaction of solitons in nonintegrable problems coincides with the scenario of the standard interaction of solitons of the KdV equation . The same problem for integrable versions of KdV was previously solved by the inverse scattering method. In addition to describing the interaction scenario in the framework of the inverse scattering method, the shift of the trajectory of the interacting solitons was calculated . The weak asymptotic method , in the previous version , did not allow the trajectory shift to be calculated. From the point of view of the Maslow – Whitham method, this shift of trajectories is of the first order of smallness relative to the paths themselves, and this order of smallness was not taken into account earlier in the framework of the method of weak asymptotics. Therefore, the problem arose to correct the definition of a weak asymptotic soliton solution of the KdV equation so that the limiting problem included not only the main part of the trajectory, but also the first correction. As mentioned above, such a limiting problem was obtained earlier within the framework of the Maslov – Whitham method . In the paper, such a refinement of the definition of a weak solution was given. Based on this refinement, a limit problem was obtained, which includes equations from the previous limit problem and a new equation for the first correction to the soliton trajectory.