WEAK CONVERGENCE FOR NON-UNIFORM φ-MIXING RANDOM FIELDS

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SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION

In this paper, the translation of the Lax pairs of the Levi equations is presented. Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally, the involutive solutions of the Levi equations are presented.     

谱问题、发展方程族及Lax表示的等价性

讨论了四对谱问题、相应的保谱方程族以及它们Ｌａｘ表示的等价住，并利用这种等价关系可导出一些方程的Ｌａｘ表示．     

Backlund Transformations for the Isospectral and Non-isospectral Matrix Kdv Hierarchies

By transforming the usual Lax pairs of isospectral and non-isospectral matrix Kdv hierarchies into Lax pairs Riccati form, a unified explicit from of Backlund transformations and superposition formulas for these two kinds of hierarchies of equations can be ohtaincd.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

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.     

Integrable decompositions for the (2+1)-dimensional Gardner equation

In this paper, with the computerized symbolic computation, the nonlinearization technique of Lax pairs is applied to find the integrable decompositions for the (2+1)-dimensional Gardner [(2+1)-DG] equation. First, the mono-nonlinearization leads a single Lax pair of the (2+1)-DG equation to a generalized Burgers hierarchy which is linearizable via the Hopf–Cole transformation. Second, by the binary nonlinearization of two symmetry Lax pairs, the (2+1)-DG equation is decomposed into the generalized coupled mixed derivative nonlinear Schrödinger (CMDNLS) system and its third-order extension. Furthermore, the Lax representation and Darboux transformation for the CMDNLS and third-order CMDNLS systems are constructed. Based on the two integrable decompositions, the resonant N-shock-wave solution and an upside-down bell-shaped solitary-wave solution are obtained and the relevant propagation characteristics are discussed through the graphical analysis.     

A family of integrable differential–difference equations,its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs

A family of integrable differential–difference equations is derived by the method of Lax pairs. A discrete Hamiltonian operator involving two arbitrary real parameters is introduced. When the parameters are suitably selected, a pair of discrete Hamiltonian operators is presented. Bi-Hamiltonian structure of obtained family is established by discrete trace identity. Then, Liouville integrability for the obtained family is proved. Ultimately, through the binary nonlinearization of the Lax pairs and adjoint Lax pairs, every differential–difference equation in obtained family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

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.     

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.     

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.     

Stäckel transform of Lax equations

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.     

Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations

The Lax integrability of the coupled KdV equations derived from two-layer fluids [S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids, J. Phys. A: Math. Gen. 39 (2006) 513-527] is investigated by means of prolongation technique. As a result, the Lax pairs of some Painlevé integrable coupled KdV equations and several new coupled KdV equations are obtained. Finally, the Miura transformations and some coupled modified KdV equations associated with the Lax integrable coupled KdV equations are derived by an easy way.     

Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg–de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica.     

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.     

差分方程拉克斯对的一种构造性方法

非交换微分在讨论数学物理中的偏微分方程时起着十分重要的作用.最近,作者利用一个具体的非交换外微分建立了一种求差分微分方程拉克斯对的方法,由此检验了该方程的可积性.本文给出了讨论全差分方程的对应理论.另外还讨论了一个格子形变的KdV(LMKdV)方程,并求得了它的拉克斯对.     

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.     

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.     

Lax Pairs for Linear Hamiltonian Systems

Siberian Mathematical Journal - We construct Lax pairs for linear Hamiltonian systems of differential equations. In particular, the Gröbner bases are used for computations. It is proved that...     

Symmetry reductions of the Lax pair for the 2 + 1-dimensional Konopelchenko–Dubrovsky equation

Based on the symbolic computational system – Maple, the similarity reduction arising from the classical Lie point symmetries of the Lax pair for the 2 + 1-dimensional Konopelchenko–Dubrovsky (KD) equation, is carried out. We obtain several interesting reductions. By analyzing not only the reduced Lax pair but also the KD equation reduced under the same symmetry group, we find that the reduced Lax pairs do not always lead to the reduced KD equation.