Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation (original) (raw)
Related papers
The Bi-Hamiltonian Structure of the Perturbation Equations of KdV Hierarchy
Eprint Arxiv Solv Int 9601003, 1996
The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions.
Methods and Applications of Analysis, 2000
A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local 2 + 1 dimensional bi-Hamiltonian formulations and consequent 2 + 1 dimensional hereditary recursion operators.
The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy
Physics Letters A, 1996
The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions.
Matrix KdV and mKdV hierarchies: Noncommutative soliton solutions and explicit formulae
2009
Soliton-like solutions to the ordinary Schrödinger equation within standard quantum mechanics J. Math. Phys. 53, 052102 (2012) Integrability of nonlinear wave equations and solvability of their initial value problem J. Math. Phys. 53, 043701 (2012) Spontaneous soliton generation in the higher order Korteweg-de Vries equations on the half-line Chaos 22, 013138 (2012) The quasi-periodic solutions of mixed KdV equations J. Math. Phys. 53, 033508 (2012) Soliton interactions in some semidiscrete integrable systems
Soliton solutions for the super mKdV and sinh-Gordon hierarchy
2006
The dressing and vertex operator formalism is emploied to study the soliton solutions of the N = 1 super mKdV and sinh-Gordon models. Explicit two and four vertex solutions are constructed. The relation between the soliton solutions of both models is verified.
Applied Mathematics and Computation, 2018
We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2 , R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy and shows its Liouville integrability. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high-order symmetries.
Integrable counterparts of the D-Kaup–Newell soliton hierarchy
Applied Mathematics and Computation, 2014
Two integrable counterparts of the D-Kaup-Newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal Lie algebra soð3; RÞ. An application of the trace identity presents Hamiltonian or quasi-Hamiltonian structures of the resulting counterpart soliton hierarchies, thereby showing their Liouville integrability, i.e., the existence of infinitely many commuting symmetries and conserved densities. The involved Hamiltonian and quasi-Hamiltonian properties are shown by computer algebra systems.
An integrable generalization of the Kaup–Newell soliton hierarchy
Physica Scripta, 2014
A generalization of the Kaup-Newell spectral problem associated with sl (2,) is introduced and the corresponding generalized Kaup-Newell hierarchy of soliton equations is generated. Bi-Hamiltonian structures of the resulting soliton hierarchy, leading to a common recursion operator, are furnished by using the trace identity, and thus, the Liouville integrability is shown for all systems in the new generalized soliton hierarchy. The involved bi-Hamiltonian property is explored by using the computer algebra system Maple.
2018
We present an enlarged spectral problem starting from a generalization of the D-Kaup-Newell (D-KN) spectral problem. Then we solve the enlarged zerocurvature equations to produce a series of Lax pairs and corresponding evolution equations formulating the desired integrable couplings. A reduction is made of the original enlarged spectral problem and the related zero-curvature equations are solved generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original enlarged spectral problem of a generalized D-KN hierarchy to furnish Hamiltonian structures. Then we apply the variational identity to the reduced problem to see its bi-Hamiltonian structures. Both hierarchies have infinitely many commuting symmetries and conserved densities, i.e., are Liouville integrable.