Hodograph solutions for the Manakov–Santini equation

Abstract

We investigate the integrable (2+1)-dimensional Manakov–Santini equation from the Lax–Sato form. Several particular two- and three-component reductions are considered so that the Manakov–Santini equation can be reduced to systems of hydrodynamic type. Then one can construct infinitely many exact solutions of the equation by the hodograph method.

Dispersionless BKP Equation, the Manakov–Santini System and Einstein–Weyl Structures

“…It is possible to obtain condition (12) by means of a reduction in the MS hierarchy, characterised by the existence of wave function of adjoint inear operators of the hierarchy with special analytic properties (with respect to the spectral variable). The technique for constructing this type of reduction was developed in [9].…”

“…Let us consider a multicomponent generalisation of reduction (18) ψ = ∏(λ − η i ) α i and complement it with a standard waterbag ansatz [11] for the wave functions of MS inear operators (9) ψ = λ + ∑ γ j ln(λ − φ j ). Some special examples of related reductions were considered in [12]. These reductions lead to (1+1)-dimensional systems of hydrodynamic type, defining the dynamics with respect to y,…”

Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities

“…The Manakov-Santini system possesses a non-Hamiltonian Lax pair and the construction of related hierarchy [15] within the Lax-Sato formalism [6,8] unifies two original approaches based on different underlying structures: first by Takasaki and Takebe [33,34] and the second one by Martínez Alonso and Shabat [21,22,23]. The Manakov-Santini hierarchy and its generalizations were further studied in several works, see for instance [6,7,8,9,10,25].…”

“…The aim of this work is an extension of the Lax-Sato formalism of Manakov-Santini hierarchy to a more general class of integrable systems, in particular such as the dispersionless modified KP equation or the so-called r-th systems [2,3,4]. Influenced by the papers [7,8,9,10] we generalize the Lax-Sato formalism of Manakov-Santini hierarchy by means of the Lax hierarchy (2.1), where two linear operators P and R are introduced. In Theorem 1 we find the conditions, on the operators P and R, for the mutual commutativity of equations from the hierarchy (2.1).…”

1 + 1 spectral problems arising from the Manakov–Santini system

“…is a member of the Manakov-Santini hierarchy [10] and it is well known that it has several interesting reductions [3], [4], [12].…”