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A hierarchy of integrable partial differential equations in $2{+}1$ dimensions associated with one-parameter families of one-dimensional vector fields

S. V. Manakov^a, P. M. Santini<sup>b

a</sup> L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
^b University of Rome "La Sapienza"

Abstract: We introduce a hierarchy of integrable partial differential equations in $2+1$ dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions.

Keywords: integrable system, inverse scattering transform, inverse spectral transformation, family of vector fields, nonlinear Riemann problem.

DOI: 10.4213/tmf6076

 English version: Theoretical and Mathematical Physics, 2007, 152:1, 1004–1011

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