Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation (original) (raw)
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.
An integrable counterpart of the D-AKNS soliton hierarchy from
Physics Letters A, 2014
An integrable counterpart of the D-AKNS soliton hierarchy is generated from a matrix spectral problem associated with so(3, R). Hamiltonian structures of the resulting counterpart soliton hierarchy are furnished by using the trace identity, which yields its Liouville integrability.
Nonautonomous mixed mKdV–sinh–Gordon hierarchy
2010
The construction of a nonautonomous mixed mKdV/sine-Gordon model is proposed by employing an infinite dimensional affine Lie algebraic structure within the zero curvature representation. A systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time dependent functions. A particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mKdV system into a pure sine-Gordon soliton.
Soliton equations related to the affine Kac-Moody algebra D 4 (1)
The European Physical Journal Plus
We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D (1) 4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions
2000
We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sGmKdV hierarchy, including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebrogeometric solutions. Although we mainly focus on sG-type equations, our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.
Negative even grade mKdV hierarchy and its soliton solutions
2009
In this paper we discuss the algebraic construction of the mKdV hierarchy in terms of an affine Lie algebraŝl(2). An interesting novelty araises from the negative even grade sector of the affine algebra leading to nonlinear integro-differential equations admiting non-trivial vacuum configuration. These solitons solutions are constructed systematically from generalization of the dressing method based on non zero vacua. The sub-hierarchies admiting such class of solutions are classified.
Soliton equations related to the affine Kac-Moody algebra D^(1)_4
arXiv (Cornell University), 2014
We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D (1) 4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
Open Physics, 2021
This article studies the fifth-order KdV (5KdV) hierarchy integrable equation, which arises naturally in the modeling of numerous wave phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves, and magneto-sound propagation in plasma. Two innovative integration norms, namely, the G G 2 () ′-expansion and ansatz approaches, are used to secure the exact soliton solutions of the 5KdV type equations in the shapes of hyperbolic, singular, singular periodic, shock, shock-singular, solitary wave, and rational solutions. The constraint conditions of the achieved solutions are also presented. Besides, by selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in three-dimensional, two-dimensional, and contour graphs. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex nonlinear phenomena.
Solitons from dressing in an algebraic approach to the constrained KP heirachy
Journal of Physics A: Mathematical and General, 1998
The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1 + 1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-Bäcklund Wronskian solutions of the constrained KP hierarchy.
Journal of Mathematical Physics, 2011
The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold [``Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,'' J. Math. Phys. 50, 073510 (2009)]. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is
A class of soliton solutions for the N= 2 super mKdV/Sinh-Gordon hierarchy
2008
Employing the Hirota's method, a class of soliton solutions for the N = 2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N = 1 supersymmetric mKdV equations connected by nontrivial algebraic identities. Using the super Miura transformation, we obtain solutions of the N = 2 super KdV equations. These are shown to generalize solutions derived previously. By using the mKdV/sinh-Gordon hierarchy properties we generate the solutions of the N = 2 super sinh-Gordon as well.