The matrix Lax representation of the generalized Riemann equations and its conservation Laws
📝 Abstract
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
💡 Analysis
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
📄 Content
arXiv:1106.1274v2 [nlin.SI] 4 Jul 2011 The matrix Lax representation of the generalized Riemann equations and its conservation laws November 14, 2018 Ziemowit Popowicz Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-205, Wrocław, Poland tel. 48-71-375-9353, fax 48-71-321-4454 ziemek@ift.uni.wroc.pl Abstract It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained. Introduction. The theory of the hydrodynamical type systems of the non-linear equations [1] , integrable by the generalized hodograph method [2] is closely related to the over- determined systems of first order partial differential linear equations . It is achieved introducing the so called Riemann invariants in which the hydrodynamic type system is rewritten in the diagonal form as ri t = µ(r)iri x where i = 1, 2 . . . , r = (r1, r2…rN) and no summation on the repeated indices. Thus, it is a linear systems of first order partial differential equations with variable coefficients µ(r). When N = 1 the equation on Riemann invariant reduces to the so called Riemann equation rt = −rrx which have been investigated in many papers and could be considered as the dispersionless limit of the Korteweg de Vries equation [3]. Recently the interesting generalization of the Riemann equation to the multicomponent case (∂/∂t + u∂/∂x)Nu = 0, N = 1, 2 . . . have been proposed in [4, 5, 6] When N = 2 this generalized system is reduced to the Gurevich-Zybin system [7, 8] or to the equation which describes the non-local gas dynamic [9] or to the Whitham type system [1]. It is possible also to reduces the N = 2 generalized Riemann equation [4] to the celebrated Hunter-Saxton equation [10], sometimes referred as the Hunter-Zheng equation [11]. The Hunter-Saxton equation has been 1 studied in almost all respects, including its complete solvability by quadratures [12, 13], construction of an infinite number of conservation laws [10, 11, 14], relationship with the Camassa-Holm equation and the Liouville equation [15], Bi-Hamiltonian formulation [11, 15], integrable finite-dimensional reductions [11, 16], global solution properties [17, 18], to mention only a few of numerous publications on this equation. For an arbitrary N the investigation of the properties of the generalized Riemann equation just started in [4, 5, 6]. It was indicated that N = 3 generalized Riemann equation possess the matrix Lax representation, the Hamiltonian formulations and huge number of polynomial and non-polynomial conservation laws. However this matrix Lax representation is a free-form, because it contains one arbitrary function which could be fixed but in not a unique manner, taking into account the integra- bility condition on the Lax representation. In this paper we present the matrix Lax representation for an arbitrary N which is not a free-form. This representation for N = 2 could be reduced to the very well known energy-dependent second-order Lax operator [19, 20, 13] while for N = 3 to the energy depended third-order Lax operator introduced in [21]. Some of the func- tions, which constitute the scalar Lax pair for N = 2, 3, are also the non-polynomial conserved Hamiltonian functionals. Moreover from this matrix Lax representation it is possible to obtain the conserved densities which are explicitly time depended. We present the operator, in some sense the analogue of the recursion operator, which generates an infinite number of non-polynomial conservation laws. As the by-product of our analysis we present new series of the non-polynomial conservation laws for the Hunter-Saxton equation. The paper is organised as follows. The first section describes the generalized Riemann equation and shows its connection with the multi-component generaliza- tions of the Hunter-Saxton equation. In the second section we define new matrix representation for an arbitrary N extended Riemann equation. The third section describes the reduction of matrix Lax representation, for N = 2 and 3, to the scalar Lax representation. In the fourth section the conservation laws for N = 2, 3 general- ized Riemann equation are obtained directly from the Lax representation. The fifth section describes the algorithm of generations of the non-polynomial conservation laws. 1 Generalized Riemann Equation. The hydrodynamical Riemann equation ut = −uux, (1) have been recently generalized to the multicomponent case [4, 5, 6] as DN t u = 0, Dt = ∂/∂t + u∂/∂x, (2) where N = 1, 2, 3 . . . . We can rewrite the last equati
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