Construction blowing up solutions of modified Novikov-Veselov equation by second order Enneper surface
Keywords:
Dirac operator, modified Novikov-Veselov equation, Mutard transformation, blowing up solutions, Enneper surfaceAbstract
In this paper, we constructed blowing-up solutions of the modified Veselov-Novikov equation (which is a two-dimensionalization of modified Korteweg-de Vries equation) using inversions of the second order Enneper minimal surface. As [1] these solutions have a singularity at one point in space- time. Algorithm for solving the modified Veselov-Novikov equation was given in [2] and in [3] obtained a geometrical interpretation of Moutard transformation. It is given by the solution of the Dirac equation Dψ = 0 and three real constants. And any solution of this equation defines a surface in three-dimensional Euclidean space, given up accurate to translations, with the help of the Weierstrass representation. Fixing the three constants, we completely fix the surface. On this surface given a conformal parameter, and the potential U of the Dirac operator is the potential of representation of the surface.Applying inversion to this surface with center at the origin, we obtain a new surface with the same conformal parameter and new potential. It turns out that the potential of inverted surface is exactly the potential constructed by Moutard transformation from the data of [3]. In a result of this article, this potential has been built (theorem 1) for yet unknown solutions of a linear system of equations (8) by algorithm of Moutard transformations; was obtained geometric interpretation of Moutard transformations on the example of second order Enneper surface, i.e, for explicit solutions of linear system of equations found potentials (theorem 2) that satisfy the modified Veselov-Novikov equation.
References
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