The Gibbons–Tsarev equation is a second order nonlinear partial differential equation. In its simplest form, in two dimensions, it may be written as follows:
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The equation arises in the theory of dispersionless integrable systems, as the condition that solutions of the Benney moment equations may be parametrised by only finitely many of their dependent variables, in this case 2 of them. It was first introduced by John Gibbons and Serguei Tsarev in 1996, This system was also derived, as a condition that two quadratic Hamiltonians should have vanishing Poisson bracket.
Relationship to families of slit maps
The theory of this equation was subsequently developed by Gibbons and Tsarev. In
Both these equations hold for all pairs
This system has solutions parametrised by N functions of a single variable. A class of these may be constructed in terms of N-parameter families of conformal maps from a fixed domain D, normally the complex half
Analytic solution
Some examples of analytic solutions of the 2-dimensional system are:
An elementary family of solutions to the N-dimensional problem may be derived by setting:
where the real parameters
The polynomial on the right hand side has N turning points,
the