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.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 147–156, July, 2007.
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Manakov, S.V., Santini, P.M. A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields. Theor Math Phys 152, 1004–1011 (2007). https://doi.org/10.1007/s11232-007-0084-2
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DOI: https://doi.org/10.1007/s11232-007-0084-2