Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper, the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations. After this, two kinds of Darboux transformations in different directions and their mixed Darboux transformations of this hierarchy are constructed. These symmetries and Darboux transformations might be useful in GromovWitten theory of CP¹.     

Burgers and Kadomtsev-Petviashvili hierarchies: A functional representation approach

Functional representations of (matrix) Burgers and potential Kadomtsev-Petviashvili (pKP) hierarchies (and others), as well as some corresponding Bäcklund transformations, can be obtained surprisingly simply from a “discrete” functional zero-curvature equation. We use these representations to show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.     

Darboux transformations for the strict KP hierarchy

Theoretical and Mathematical Physics - We introduce the notion of Darboux transformations for the strict KP hierarchy. We previously showed that solutions of this integrable hierarchy can be...     

The dressing method and nonlocal Riemann-Hilbert problems

Summary We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.     

Monodromy-free Schrödinger operators with quadratically increasing potentials

We consider one-dimensional monodromy-free Schrödinger operators with quadratically increasing rational potentials. It is shown that all these operators can be obtained from the operator -?² + x² by finitely many rational Darboux transformations. An explicit expression is found for the corresponding potentials in terms of Hermite polynomials.     

Infinite-dimensional flag manifolds in integrable systems

In this paper, we present several instances where infinite-dimensional flag varieties and their holomorphic line bundles play a role in integrable systems. As such, we give the correspondence between flag varieties and Darboux transformations for the KP hierarchy and the nth KdV hierarchy. We construct solutions of the nth MKdV hierarchy from the space of periodic flags and we treat the geometric interpretation of the Miura transform. Finally, we show how the group extension connected with these line bundles shows up at integrable deformations of linear systems on ¹().     

Scattering of solitons in the formalism of the Darboux transform

By means of the technique of Darboux transformations formulas are obtained which describe the scattering of solitons in the Kadomtsev-Petviashvili equations; the KdV equation, the nonlocal KdV equation, the two-dimensionalized Toda lattice and its periodic reductions, and, in particular, the sine-Gordon equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 12, pp. 136–141, 1982.     

Lagrangian Chains and Canonical Bäcklund Transformations

We consider Darboux transformations for operators of arbitrary order and construct the general theory of Bäcklund transformations based on the Lagrangian formalism. The dressing chain for the Boussinesq equation and its reduction are demonstrative examples for the suggested general theory. We also discuss the well-known Bäcklund transformations for classical soliton equations.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy

The Novikov equation and a negative flow of the Novikov hierarchy are related to a negative flow of the Sawada–Kotera hierarchy and the Sawada–Kotera equation by reciprocal transformations, respectively. With the help of the Darboux transformations for the negative flow of the Sawada–Kotera hierarchy, the Sawada–Kotera equation and reciprocal transformations, we obtain a parametric representation for $N$-soliton solutions to the Novikov equation and the negative flow of the Novikov hierarchy.     

A Supersymmetric AKNS Problem and Its Darboux‐Bäcklund Transformations and Discrete Systems

In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and Bäcklund transformations are given for the supersymmetric modified Korteweg‐de Vries, sinh‐Gordon, and nonlinear Schrödinger equations. These Darboux and Bäcklund transformations are adopted for the constructions of integrable discrete super systems, and both semidiscrete and fully discrete systems are presented. Also, the continuum limits of the relevant discrete systems are worked out.     

Supersymmetric structure of Kadomtsev-Petviashvili equation: Connection between supersymmetry and Bäcklund transformations

Factorization of the Sato integrodifferential operator that gives the hierarchy of the Kadomtsev-Petviashvili equation is considered. The supersymmetric structure of this hierarchy is investigated by means of the factorization. A connection between the Miura transformations, Bäcklund transformations, and supersymmetry is established.P. N. Lebedev Physics Institute, Rossiskoi Akademii Nauk. Translated from Teorecheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 426–432, June, 1992.     

Integrable boundary conditions for (2+1)-dimensional models of mathematical physics

We consider the question of integrable boundary-value problems in the examples of the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. We discuss the problems that are integrable from the standpoints of two basic definitions of integrability. As a result, we propose a method for constructing a hierarchy of integrable boundary-value problems where the boundaries are cylindric surfaces in the space of three variables. We write explicit formulas describing wide classes of solutions of these boundary-value problems for the two-dimensional Toda chain and Kadomtsev-Petviashvili equation.     

Darboux System as Three-Dimensional Analog of Liouville Equation

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux–Laplace transforms for second-order equations with two independent variables to the case of three-dimensional linear hyperbolic equations of the third order. In this paper we construct examples of such transformations. We consider applications to the problem of orthogonal curvilinear coordinate systems in ?³.     

On soliton and periodic solutions of Maxwell–Bloch system for two-level medium with degeneracy

Maxwell–Bloch (MB) system describing the resonant propagation of electromagnetic pulses in either two-level media with degeneracy in angle moment projection or a three-level media with equal oscillator forces is considered. An inhomogeneous broadening of energy levels and a polarization of the wave are accounted. The equations are integrated by the binary Darboux transformations technique. Pulses corresponding to a transition between levels with the largest population difference are shown to be stable. The solution describing the propagation of pulses in the medium excited by a periodic wave is obtained. The hierarchy of infinitesimal symmetries is obtained by means of Darboux transformation.     

Cylindrical Kadomtsev-Petviashvili equation: Old and new results

We review results on the cylindrical Kadomtsev-Petviashvili (CKP) equation, also known as the Johnson equation. The presentation is based on our results. In particular, we show that the Lax pairs corresponding to the KP and the CKP equations are gauge equivalent. We also describe some important classes of solutions obtained using the Darboux transformation approach. We present plots of exact solutions of the CKP equation including finite-gap solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 304–320, August, 2007.     

一个新的Lie代数和它的应用

通过构造一个新的Lie代数,利用它相应的Loop代数设计等谱Lax对,根据其相容性条件,得到了一族Lax可积方程族,其一种约化形式为著名的AKNS族．根据迹恒等式得到该方程族的Hamilton结构．利用该可积方程族可以进一步研究它的达布变换、对称、代数几何解等相关性质．     

Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral transformations are used to define discrete analogs for cmc-1 surfaces in hyperbolic space and to obtain a discrete version of Bryant's Weierstrass type representation. Received: 13 September 1999 / Revised version: 27 March 2000     

Lumps and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.