一个经典的可积Neumann系统和Boussinesq族Lax对的非线性化

本文在位势与特征函数之间的Neumann约束条件下,经典Boussinesq族的Lax对被非线性化成为自然相容的Lax系统;而且,其为Liouville完全可积的Hamiltonian系统,同时获得了Boussinesq方程解的对合表示。     

NEW SYMPLECTIC MAPS： INTEGRABILITY AND LAX REPRESENTATION

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Real Lax spectrum implies spectral stability

We consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset of the real line that gives rise to stable eigenvalues is contained in the Lax spectrum. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and it maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples, both in and not in the AKNS hierarchy.     

A new extended matrix KP hierarchy and its solutions

Starting from the matrix KP hierarchy and adding a new τ_B flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times t_A and τ_B and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of (2+1)-dimensional AKNS equations with self-consistent sources and two types of Davey-Stewartson equations with selfconsistent sources.     

The Hamiltonian Systems of the LCZ Hierarchy by Nonlinearization

In this paper, we first search for the Hamiltonian structure of LCZ hierarchy by use of a trace identity. Then we determine a higher-order constraint condition between the potentials and the eigenfunctions of the LCZ spectral prob lem and under this constraint condition, the Lax pairs of LCZ hierarchy are all nonlinearized into the finite-dimensional integrable Hamiltonian systems in Liouville sense.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

HOW TO CONSTRUCT LAX REPRESENTATION FOR CONSTRAINED FLOWS OF THEBOUSSINESQ HIERARCHY VIA ADJOINT REPRESENTATIONS

1IntroductionTheBoussinesqequationarisesinseveralphysicalapplicationandhasbeenstudiedquiteextensivelyinthepast[1--3].Inarecentpaper[4],itwasfoundthattheBoussinesqhierarchycanbeobtainedfromthezero-curvatureconditionassociatedwiththegroupSL(3,R).ThisshowsadirectrelationshipbetweenthegroupSL(3,R)andtheW3algebraofZamolodchikov.Recentlytherehasbeenconsiderableinterestinthedecompositionofsolitonequationsviaconstraintsrelatingpotentialandeigenfunctions,becausethedecompositionprovidesaneffectiveme…     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg–de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.     

Positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

Commutativity of the extended KP flows

The commutativity problem of the extended KP hierarchy is analyzed. The compatibility equation of two extended KP flows is constructed, together with its Lax representations involving two extended Lax operators. The resulting theory shows that the extended KP hierarchy is a natural generalization of the KP flows, but does not commute unlike the constrained KP hierarchy. A few particular examples are computed, along with their Lax pairs.     

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

We study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.     

一个有限维完全可积系统和MDWW方程解的对合表示

本文讨论高阶ＭＤＷＷ方程的Ｌａｘ对，在位势与特征函数之间的约束条件下，Ｌａｘ系统被非线性化成为有限维Ｌｉｏｕｖｉｌｌｅ完全可积系统．并且获得了高阶ＭＤＷＷ方程解的对合表示．     

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.     

A Kind of Discretization of the KdV Hierarchy

We propose a method for introducing higher-order terms in the potential expansion in order to study the continuum limits of the Toda hierarchy. These higher-order terms are differential polynomials in the lower-order terms. This type of potential expansion allows using fewer equations in the Toda hierarchy to recover the KdV hierarchy by the so-called recombination method. We show that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend toward the corresponding objects of the KdV hierarchy in the continuum limit. This method gives a kind of discretization of the KdV hierarchy.     

On a Schwarz Lemma for Bounded Symmetric Domains

We develop a new systematic approach to the Boussinesq (Bsq) hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the Bsq hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro -geometric setting including associated Burchnall - Chaundy curves, Baker - Akhiezer functions, trace formulas, and Dubrovin - type equations for Dirichlet and Neumann divisors.     

Invariant manifolds and Lax pairs for integrable nonlinear chains

We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.     

非等谱Lax算子族的Virasoro代数

非等谱Ｌａｘ算子族的Ｖｉｒａｓｏｒｏ代数马文秀（复旦大学数学研究所，上海２００４３３）国家博士后科学基金资助项目．１９９１年７月２５日收到．１９９２年７月６日收到修改稿．一、引言Ｌａｘ算子方法［１］在可积系统理论中有着广泛的应用．从一个谱问题出发我们...