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.     

离散晶格的正族、负族及其Hamilton结构

构造了一个新的等谱问题，利用相容性条件，推导出离散晶格方程的正族和负族。再利用迹恒等式，建立其Hamilton 结构。获得的离散方程族的达布变换、双线性化、对称、守恒率及其精确解也值得进一步研究。     

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().     

Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

Summary. We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n -th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, the lump solution, and the Johnson-Thompson lump, and the corresponding determinant formulae for the nonlinear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons, we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases. Received June 4, 1997; final revision received March 6, 1998; accepted March 23, 1998     

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...     

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.     

Binary Darboux transformations andN-wave systems in rings

The covariance theorems for elementary and binary Darboux transformations in rings are formulated and proved for generalized Zakharov-Shabat problems. The definition of the elementary Darboux transformation is extended to an arbitrary number of orthogonal idempotents. The binary transformation is defined as a sequence of elementary transformations for direct and conjugate problems. The heredity property for the reduction constraints is established for some UV pairs in rings; hence, the transformation generates solutions and infinitesimal symmetries of the corresponding zero-curvature equations. The explicit expressions for the transformations, solitons, and infinitesimals are given in the general case and in physically significant cases of extended non-AbelianN-wave equations (with linear terms added). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 239–250, February, 2000.     

Darboux transformations and recursion operators for differential-difference equations

We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.     

Darboux Transformation and Exact Solutions for Two Dimensional A 2n−1 (2) Toda Equations

The Darboux transformation for the two dimensional A⁽²⁾_2n-1 Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Nonlocal Symmetries for Bilinear Equations and Their Applications

In this paper, nonlocal symmetries for the bilinear KP and bilinear BKP equations are re-studied. Two arbitrary parameters are introduced in these nonlocal symmetries by considering gauge invariance of the bilinear KP and bilinear BKP equations under the transformation     . By expanding these nonlocal symmetries in power series of each of two parameters, we have derived two types of bilinear NKP hierarchies and two types of bilinear NBKP hierarchies. An impressive observation is that bilinear positive and negative KP (NKP) and BKP hierarchies may be derived from the same nonlocal symmetries for the KP and BKP equations. Besides, as two concrete examples, we have derived bilinear Bäcklund transformations for   t ₋₂  -flow of the NKP hierarchy and   t ₋₁  -flow of the NBKP hierarchy. All these results have made it clear that more nice integrable properties would be found for these obtained NKP hierarchies and NBKP hierarchies. Because KP and BKP hierarchies have played an essential role in soliton theory, we believe that the bilinear NKP and NBKP hierarchies will have their right place in this field.     

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Backlund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.     

Darboux Transformations Associated with Boiti-Tu Eigenvalue Problem

To the hierarchy of nonlinear evolution equations which associate with Boiti-Tu eigenvalue problem, an explicit and universal form of Baeklund transformations and auniversal proof are presented. It is called Darboux transformation. By this method, to ask for a new solution of every system of equations of the hierarchy, it is sufficient to solve some linear problems. Here the constraints at the boundary for the potentials (for example, at x=\pm \infinity) are removed.     

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.     

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.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

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．     

On a vector long wave-short wave-type model

A new vector long wave-short wave-type model is proposed by resorting to the zero-curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave-short wave-type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two-component long wave-short wave-type model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena.     

Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrödinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order.