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1.
Li Li 《Physics letters. A》2011,375(3):511-516
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs. 相似文献
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3.
Following Sato's famous construction of the KP hierarchy as a hierarchy of commuting Lax equations on the algebra of microdifferential operators, it is shown that n-reduction leads to a recursive scheme for these equations. Explicit expressions for the recursion operators and the Hamiltonian operators are obtained. 相似文献
4.
In terms of the operator Nambu 3-bracket and the Lax pair (L, Bn) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, Bn, Bm). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, Bn, Bm). Furthermore we derive some no integrable evolution equations and present their single soliton solutions. 相似文献
5.
J. Chen X. Geng 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,50(3):445-452
The Kadometsev-Petviashvili (KP) and
modified KP (mKP) equations are retrieved from the first
two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy.
Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately
reduced to integrable finite-dimensional Hamiltonian
systems in view of the r-matrix theory. Finally,
the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that
some specially explicit quasi-periodic solutions to the KP and mKP equations
are synchronously given in terms of theta functions through the Jacobi
inversion. 相似文献
6.
A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations. 相似文献
7.
We construct the matrix generalization of the N=2 supersymmetric GNLS hierarchies. This is done by exhibiting the corresponding matrix super Lax operators in terms of N=2 superfields in two different superfield bases. We present the second Hamiltonian structure and discrete symmetries. We then extend our discussion by conjecturing the Lax operators of different reductions of the N=2 supersymmetric matrix KP hierarchy and discuss the simplest examples. 相似文献
8.
In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev–Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitely-many conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates. 相似文献
9.
We study Darboux–Bäcklund transformations (DBTs) for the q-deformed Korteweg–de Vries hierarchy by using the q-deformed pseudodifferential operators. The elementary DBTs are triggered by the gauge operators constructed from the (adjoint) wave functions of the associated linear systems. Iterating these elementary DBTs, we obtain not only q-deformed Wronskian-type but also binary-type representations of the tau-function of the hierarchy. 相似文献
10.
G. Carlet J. van de Leur H. Posthuma S. Shadrin 《Communications in Mathematical Physics》2014,326(3):815-849
To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in Buryak et al. (J Differ Geom 92:153–185, 2012), Buryak et al. (J Geom Phys 62:1639–1651, 2012) to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action. 相似文献
11.
《Journal of Nonlinear Mathematical Physics》2013,20(2):193-204
Abstract For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the symmetry constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations. 相似文献
12.
With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element (1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions, the mKdV equation and a generalized linear KdV equation. The algebro-geometric solutions (also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced. 相似文献
13.
Luen-Chau Li 《Communications in Mathematical Physics》1999,203(3):573-592
Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian
formulation of Lax equations. Examples for which our formalism applies include the Benny hierarchy, the dispersionless Toda
lattice hierarchy, the dispersionless KP and modified KP hierarchies, the dispersionless Dym hierarchy, etc.
Received: 10 February 1998 / Accepted: 9 December 1998 相似文献
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The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov–Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of c=1 string theory except that the Orlov–Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermion bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so-called Lambert curve emerges in a specialization of its solution. This seems to be another way of deriving the spectral curve of the random matrix approach to Hurwitz numbers. 相似文献
16.
Maarten Bergvelt Michael Gekhtman Alex Kasman 《Mathematical Physics, Analysis and Geometry》2009,12(2):181-200
Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these
operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the
case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to
the bispectrality of (scalar) differential operators.
相似文献
17.
This paper is devoted to the study of the underlying linearities of the coupled
Harry--Dym (cHD) soliton hierarchy, including the well-known cHD equation. Resorting
to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian
systems associated with soliton equations are presented, constituting the
decomposition of the cHD soliton hierarchy. After suitably introducing the
Abel--Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be
ultimately reduced to linear superpositions,
expressed by the Abel--Jacobi variables. 相似文献
18.
L. A. Dickey 《Communications in Mathematical Physics》1995,167(1):227-233
A short proof is given to the fact that the additional symmetries of the KP hierarchy defined by their action on pseudodifferential operators according to Fuchssteiner-Chen-Lee-Lin-Orlov-Shulman coincide with those defined by their action on -functions as Sato's Bäcklund transformations. A new simple formula for the generator of additional symmetries is also presented. 相似文献
19.
Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W
1 + symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow
1 + symmetries of the dispersionless hierarchy through their action on the tau function. 相似文献
20.
Raúl Felipe 《Reports on Mathematical Physics》2003,52(2):167-176
A new integrable hierarchy, with equations defined by double brackets of two matrix pseudo-differential operators (Lax pairs), is constructed. Some algebraic properties are demonstrated. It is also shown that each equation is equivalent to a certain gradient flow. A new version of the Zakharov-Shabat type equations is proved. Formal solutions of this hierarchy are constructed using a matrix “double bracket bilinear identity”. 相似文献