Integrability structures of the generalized Hunter–Saxton equation

Abstract

We consider integrability structures of the generalized Hunter–Saxton equation. We obtain the Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give examples of employing the higher order symmetries to constructing exact globally defined solutions for the generalized Hunter–Saxton equation.

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

“…The symmetry reduction of equation ( 22) w.r.t. u y = 0 coincides with the generalized Hunter-Saxton equation [8,9,3,24] u tx = u u xx + β u 2…”

“…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 .…”

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

“…The reciprocal transformations [24] , Homotopy decomposition method [25] , bivariate generalized fractional order of the Chebyshev functions (BGFCF) [26] , cubic trigonometric B-Spline collocation method [27] , collocation method [28] , Harr wavelet quasilinearization approach [29] , and Lipschitz metric [30] , time marching scheme [31] are applied to study the diffusion of neumatic LCs. The generalized Hunter-Saxton equation is considered using integrability structures [32] , Numerical solutions of HSE using Laguerre wavelet and by using efficient approach on time domains is presented in [33] , [34] , [35] .…”