Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein–Weyl structures and the Gibbons–Tsarev equation

Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

“…For some values of N and ε the Lie algebras qN,ε admit further right extensions. In [19,20,21,22] we have shown examples of integrable pdes whose Lax representations can be inferred from such extensions with N ≥ 3. In the next two sections we consider integrable pdes that are related to q1,−1 and q1,−2 .…”

“…x with the special value β = −1 of parameter β. Hence (22) can be considered as a three-dimensional generalization of equation u tx = u u xx − u 2 x . Likewise to Section 4, equation ( 23) can be included in the infinite-component Lax representation…”

“…In particular, we prove that the known Lax representations for the potential Khokhlov-Zabolotskaya equation, the hyper-CR equation of Einstein-Weyl structures, their associated integrable hierarchies as well as some new Lax representations can be derived via a series of extensions applied to the deformations q N,ε of the family of Lie algebras R N [s] ⊗ w with N ≥ 3, where R N [s] is the commutative associative algebra of truncated polynomials of degreee N and w is the Lie algebra of vector fields on R, see Section 3 for definition of q N,ε . In Sections 4, 5, and 6 we present three equations ( 14), (22), and (27) with Lax representations generated by extensions of the Lie algebras q 1,−1 , q 1,−2 , and q 2,−1 , respectively. Equation (22) can be considered as a 3D generalization of the generalized 2D Hunter-Saxton equation [8,9,3,24] with the special value of the parameter.…”

“…In Sections 4, 5, and 6 we present three equations ( 14), (22), and (27) with Lax representations generated by extensions of the Lie algebras q 1,−1 , q 1,−2 , and q 2,−1 , respectively. Equation (22) can be considered as a 3D generalization of the generalized 2D Hunter-Saxton equation [8,9,3,24] with the special value of the parameter. In Section 7 we provide an example of the integrable pde (32) with the Lax representation defined by extensions of the Lie algebra h ⊕ w, where h is the Lie algebra of Hamiltonian vector fields on R 2 .…”