Baldomero Valiño Alonso

(La Habana, Cuba)

Infinitely narrow soliton in shallow water equations
(joint with Amaury Álvarez Cruz, Instituto de Oceanología, Cuba and Alex Méril, Université des Antilles et Guyana, Guadeloupe, France)

V.P.Maslov have shown that infinitely narrow solitons appear in Burgers’s equation as limit cases of singular solutions of Korteveg-De Fries’s equation. In this work we seek infinitely narrow solitons for shallow water equations, in the case when the profile of the depth has an isolated singular point. Following V. P. Maslov, we look for that singular solution as an asymptotical development with smooth coefficients in an appropriate algebra of generalized functions. Singular solutions of hyperbolic systems of partial differential equations in general position may have only a finite number of structuraly stable and self similar singular solutions. When the time evolves, the singular solution preserves the shape and structure of the initial condition. According to Maslov, the coeffcients of the formal asymptotic expansion of the singular solution satisfy an infinite chain of ordinary differential equations (the Hugoniot-Maslov chain corresponding to the singular solution involved). The truncation of the chain makes possible its aproximate calculation. From the point of view of numerical calculation, this approach is advantageous because it reduces the problem of solving numerically the nonlinear system of partial di¤erential equations to solving a closed system of ordinary differential equations. We interpret "infinitely narros solitons" as examples of simplified generalized functions in the sense of J. F. Colombeau and we use the rules of Colombeau’s differential algebras to do the calculations.