Nonlinearization Of Lax Pairs Research Articles

Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy

A uniform construction of quasi-periodic solutions to the negative-order Jaulent–Miodek (nJM) hierarchy is presented by using a family of backward Neumann type systems. From the backward Lenard gradients, the nJM hierarchy is put into the zero-curvature setting and the bi-Hamiltonian structure displaying its integrability. The nonlinearization of Lax pair is generalized to the nJM hierarchy such that it can be reduced to a sequence of backward Neumann type systems, whose involutive solutions yield finite parametric solutions of the nJM hierarchy. The negative [Formula: see text]-order stationary JM equation is given to specify a finite-dimensional invariant subspace for the nJM flows. With a spectral curve determined by the Lax matrix, the nJM flows are linearized on the Jacobi variety of a Riemann surface. Finally, the Riemann–Jacobi inversion is applied to Abel–Jacobi solutions of the nJM flows, by which some quasi-periodic solutions are obtained for the nJM hierarchy.

Two kinds of finite-dimensional integrable reduction to the Harry–Dym hierarchy

In this paper, two kinds of finite-dimensional integrable reduction are studied for the Harry–Dym (HD) hierarchy. From the nonlinearization of Lax pair, the HD hierarchy is reduced to a class of finite-dimensional Hamiltonian systems (FDHSs) in view of a Bargmann map and a set of Neumann type systems by a Neumann map, which separate temporal and spatial variables on the symplectic space [Formula: see text] and the tangent bundle of ellipsoid [Formula: see text], respectively. It turns out that involutive solutions of the resulted finite-dimensional integrable systems (FDISs) directly give rise to finite parametric solutions of HD hierarchy through the Bargmann and Neumann maps. The finite-gap potential to the high-order stationary HD equation is obtained that cuts out a finite-dimensional invariant subspace for the HD flows. Finally, some comparisons of two kinds of integrable reductions are then discussed.

Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super NLS-MKdV hierarchy. Under the obtained symmetry constraint, the n-th flow of the super NLS-MKdV hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R4N｜2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω2). Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω2) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.

Algebro-Geometric Solution to Two New (2+1)-DimensionalModified Kadomtsev–Petviashvili Equations

Two new (2+1)-dimensional modified Kadomtsev–Petviashvili (mKP)equations are presented, which are related to a hierarchy of(1+1)-dimensional soliton equations. Through the nonlinearization of Laxpair and the Riemann–Jacobi inversion technique, the algebro-geometricsolutions of both the mKP equations are obtained.

On the linearization of the coupled Harry–Dym soliton hierarchy

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.

Decomposition to the modified Jaulent–Miodek hierarchy

Abstract From the nonlinearization of Lax pairs (NLP), a family of finite-dimensional Hamiltonian systems (FDHSs) are presented constituting the decomposition of the modified Jaulent–Miodek (mJM) hierarchy. These FDHSs are further proved to be completely integrable in the Liouville sense in view of the generating function method. Finally, the relation between soliton equations and FDHSs is established with the aid of a set of polynomial integrals.

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

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.

Reductions to Korteweg-de Vries Soliton Hierarchy

Based on the nonlinearization of Lax pairs, the Korteweg-de Vries (KdV)soliton hierarchy is decomposed into a family of finite-dimensionalHamiltonian systems, whose Liouvilleintegrability is proved by means of the elliptic coordinates. By applying theAbel–Jacobi coordinates on a Riemann surface of hyperellipticcurve, the resulting Hamiltonian flows as well as the KdV solitonhierarchy are ultimately reduced into linear superpositions, expressedby the Abel–Jacobi variables.

Algebro-Geometric Solution to the Modified Kadomtsev–Petviashvili Equation

The modified Kadomtsev–Petviashvili (mKP) equation is split into two soliton equations in the modified Jaulent–Miodek (mJM) hierarchy, and further into integrable finite-dimensional Hamiltonian systems (FDHSs) using the nonlinearization of Lax pairs. The Abel–Jacobi coordinate is introduced to linearize the mKP flow such that its solution is reduced as a linear superposition, expressed in the Abel–Jacobi variable. Finally, the algebro-geometric solution of the mKP equation is given via the Jacobi inversion.

Relation among nonlinear evolution equations and their reductions

Abstract The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.

Lax representation and dynamical r-matrix for a new Neumann type integrable model

Abstract A new Neumann type finite-dimensional Hamilton system (FDHS) is presented by means of the nonlinearization of Lax pairs (NLP). Its Lax representation is deduced from the auxiliary spectral problem. Furthermore, it is displayed that the Lax representation satisfies a dynamic r -matrix relation in the Dirac bracket. Additionally the Lax representation and r -matrix for the Bargmann type FDHS are also given in the standard Poisson bracket. Consequently, the integrability in the Liouville sense of the resulting FDHSs is completed in the framework of r -matrix theory.

Classical r-matrix structures of integrable mappings related to the Volterra lattice

Lax representations and corresponding r-matrices for a Neumann type and a Bargmann type mappings that are generated through nonlinearization of Lax pair from Volterra lattice are presented, and thus, invariants of mappings are obtained and the integrability is established. In addition, the corresponding continuous Hamiltonian systems of invariants of these two mappings are investigated. In the case of Neumann type mapping, the corresponding continuous Hamiltonian flows are shown to be a kind of realizations of the Gaudin magnet with boundary.

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a n ,b n )T={f(ϕ)} n between the reflectionless potentials and the eigenfunctions of the general discrete Schrodinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.

The coupled HD hierarchy and a classical integrable system of the complex form

The coupled Harry Dym (HD) equations are discussed by means of the complex form of the real involutive system. Using the nonlinearization of Lax pairs of the coupled HD equation, a finite-dimensional completely integrable system in the Liouville sense is obtained. By making use of the solutions of commutative flows, the representation of the solutions for the hierarchy of the HD equation are generated.