Two-dimensional rational solitons and their blow-up via the Moutard transformation

Abstract

By using the Moutard transformation of two-dimensional Schroedinger operators we derive a procedure for constructing explicit examples of such operators with rational fast decaying potentials and degenerate $L_2$-kernels (this construction was sketched in arXiv:0706.3595) and show that if we take some of these potentials as the Cauchy data for the Novikov-Veselov equation (a two-dimensional version of the Korteweg-de Vries equation), then the corresponding solutions blow up in a finite time

arXiv comment22 pages, PDFLatex, 9 figures. v2: some computations corrected

References in corpus (2)

• Darboux Transformation and Variable Separation Approach: the Nizhnik-Novikov-Veselov equation
• Two-dimensional Schroedinger operators with fast decaying potential and multidimensional $L_2$-kernel

Cited by in corpus (10)

• Exceptional circles of radial potentials
• The exactly solvable two-dimensional stationary Schrödinger operators obtaining by the nonlocal Darboux transformation
• The Gould-Hopper Polynomials in the Novikov-Veselov equation
• Darboux-Moutard transformations and Poincare-Steklov operators
• On numerical study of the discrete spectrum of a two-dimensional Schrodinger operator with soliton potential
• The Moutard transformation and two-dimensional multi-point delta-type potentials
• Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation
• On a formation of singularities of solutions to soliton equations represented by L,A,B-triples
• On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations
• Low regularity local well-posedness for the zero energy Novikov-Veselov equation