A Relationship Between Rational and Multi-Soliton Solutions of the BKP Hierarchy (original) (raw)
Related papers
An elementary approach to the rational solutions of the KP hierarchy
We construct in a fairly elementary and simple way the rational solutions of the KP hierarchy. Our starting point is a geometric approach to soliton equations, which relies on the concept of a bihamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial T-functions of the KP hierarchy. This formula, though known in the literature, is obtained in a very direct way.
Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy
Symmetry, Integrability and Geometry: Methods and Applications, 2021
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the l-th generalized KdV hierarchy with l ≥ 2 is related to the rational space curves associated with the numerical semigroup l, lm+1,. .. , lm+k , where m ≥ 1 and 1 ≤ k ≤ l−1. We also calculate the Schur polynomial expansions of the τ-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.
An elementary approach to the polynomial τ-functions of the KP Hierarchy
Theoretical and Mathematical Physics, 2000
We give an elementary construction of the solutions of the KP hierarchy associated with polynomial Tfunctions starting with a geometric approach to soliton equations based on the conceptof a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial T-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.
The KP Theory Revisited. IV. KP Equations, Dual KP Equations, Baker–Akhiezer and Τ Functions
2009
This is the final part of a study of the theory of soliton equations and of KP equations from the standpoint of Hamiltonian mechanics. It can be read independently of the previous papers, albeit with a certain loss of the motivations which suggested the present approach. The essential idea is that the KP theory is the study of two momentum mappings associated with the Gel’fand–Dickey equations. The paper suggests an axiomatic formulation of the theory of these mappings, in the space of Laurent series with coefficients which are scalar–valued periodic functions. The KP equations treated in this paper are different from the usual Sato KP equations, but they may be connected with Sato equations by means of a suitable transformation shown at the end of the paper. Furthermore, we introduce a second set of equations, called dual KP equations, and we prove that the τ–function may be introduced as the Kähler potential of a Kähler metric associated with these equations. RUNNING TITLE: The KP...
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.
Trigonal curves and algebro-geometric solutions to soliton hierarchies I
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017
This is the first part of a study, consisting of two parts, on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, explore general properties of meromorphic functions defined as ratios of the Baker–Akhiezer functions, and determine zeros and poles of the Baker–Akhiezer functions and their Dubrovin-type equations. We analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
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.
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.
International Journal of Modern …, 1997
This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKP r,m (r, m ≥ 1) of the primary continuum integrable system-the Kadomtsev-Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKP r,m structure is a generalization of the familiar r-reduction of the full KP hierarchy to the SL(r) generalized KdV hierarchy cKP r,0. The important feature of cKP r,m hierarchies is the presence of a discrete symmetry structure generated by successive Darboux-Bäcklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKP r,1 defines a generalized 2-dimensional Toda lattice structure. Furthermore, we consider the class of truncated KP hierarchies (i.e., those defined via Wilson-Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of cKP r,m hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models. The next important step involves the reformulation of the familiar non-isospectral additional symmetries of the full KP hierarchy so that their action on cKP r,m hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the cKP r,m DB orbits. The above technical arsenal is subsequently applied to obtain complete solutions of the discrete multi-matrix models. The key ingredient is our identification of q-matrix models as DB orbits of cKP r,1 integrable hierarchies where r = (p q − 1). .. (p 2 − 1) with p 1 ,. .. , p q indicating the orders of the corresponding random matrix potentials. Applying the notions of additional symmetry structure and the technique of equivalent hierarchies turns out to be instrumental in implementing the string equation and finding closed expressions for the partition functions of the discrete multi-matrix models. As a byproduct, we obtain a representation of the τ-function of the most general DB orbit of cKP 1,1 hierarchy in terms of a new generalized matrix model.
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.