Classification of integrable hydrodynamic chains

Abstract

Using the method of hydrodynamic reductions, we find all integrable infinite (1+1)-dimensional hydrodynamic-type chains of shift one. A class of integrable infinite (2+1)-dimensional hydrodynamic-type chains is constructed.

Non-homogeneous Systems of Hydrodynamic Type Possessing Lax Representations

“…In particular, system (1.3) has only a few local infinitesimal symmetries and therefore the symmetry approach to classification of integrable 1+1-dimensional systems [7] is not applicable for systems similar to (1.3). It turns out that this system possesses infinite non-commutative hierarchy of nonlocal symmetries explicitly depending on x and t. These symmetries look like symmetries of the so called Gibbons-Tsarev systems found in [5].…”

“…There are two different possibilities: Case A: S ′′′′ = 0 and Case B: S ′′′′ = 0. In Case B the determinant of the system of linear equations (5.65) for S (4) , S (5) , S (6) should be zero. This leads to a fourth order ODE for R, whose solution is R = W 2 /W 1 , where W 2 and W 1 are arbitrary polynomials of degree 2 and 1, correspondingly.…”

On symmetries of the Gibbons–Tsarev equation

“…Systematic computation of nonlocal symmetries soon reveals that infinitely many nonlocal symmetries can be obtained through commutation. This was first observed for the unreduced Benney system in [18], where five symmetries were written out explicitly and the structure of the symmetry algebra was revealed.…”

Multi‐dimensional conservation laws and integrable systems

“…The theory of integrable commuting two‐dimensional hydrodynamic chains (see Refs. ) is related to the theory of integrable three‐dimensional quasilinear equations of second order. Indeed, taking into account the first equations one can introduce the potential function η 0 such that Then the inverse transformation is …”

“…The theory of integrable commuting two-dimensional hydrodynamic chains (see Refs. [1][2][3][4][5] = , ( 0 , 1 , … , + ), , = 0, 1, … ,…”