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Register a new userTwo binary Darboux transformations for the KdV hierarchy with self-consistent sources
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Yunbo Zeng Tsinghuan University
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Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-B{a}cklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.
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The $hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $hat B_n^{(1)}$, the Drinfeld-Sokolov $hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) use loop group factorization to construct Darboux transforms (DTs) for the $hat B_n^{(1)}$-hierarchy, (2) give a Permutability formula and scaling transform for these DTs, (3) use DTs of the $hat B_{n}^{(1)}$-hierarchy to construct DTs for the $hat B_n^{(1)}$-KdV and the isotropic curve flows of B-type, (4) give algorithm to construct soliton solutions and write down explicit soliton solutions for the third $hat B_1^{(1}$-KdV, $hat B_2^{(1)}$-KdV flows and isotropic curve flows on $mathbb{R}^{2,1}$ and $mathbb{R}^{3,2}$ of B-type.
We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noethers theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove the following results for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are differential polynomials of the formal inverse scattering solution u_f. Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The natural Virasoro action on ln(tau_f) constructed in the second paper is given by partial differential operators in ln(tau_f). (3) There is a bijection between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD_n hierarchy under the bijection.
Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with self-consistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.
It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.
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