Gibbons–Tsarev equation
The Gibbons–Tsarev equation is an integrable second order nonlinear partial differential equation. In its simplest form, in two dimensions, it may be written as follows: 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 formula_2 independent variables, one looks for solutions of the Benney hierarchy in which only formula_2 of the moments formula_4 are independent. The resulting system may always be put in Riemann invariant form. Taking the characteristic speeds to be formula_5 and the corresponding Riemann invariants to be formula_6, they are related to the zeroth moment formula_7 by: Both these equations hold for all pairs formula_10. 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 formula_11-plane, to a similar domain in the formula_12-plane but with N slits. Each slit is taken along a fixed curve with one end fixed on the boundary of formula_13 and one variable end point formula_6; the preimage of formula_6 is formula_5. The system can then be understood as the consistency condition between the set of N Loewner equations describing the growth of each slit: Analytic solution. An elementary family of solutions to the N-dimensional problem may be derived by setting: where the  real parameters formula_19 satisfy: The polynomial on the right hand side has N turning points, formula_21, with corresponding formula_22. With the formula_5and formula_7 satisfy the N-dimensional Gibbons–Tsarev equations.