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1.
By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense. 相似文献
2.
A discrete matrix spectral problem and the associated hierarchy of
Lax integrable lattice equations are presented, and it is shown that
the resulting Lax integrable lattice equations are all
Liouville integrable discrete Hamiltonian systems. A new integrable
symplectic map is given by binary Bargmann constraint of the resulting
hierarchy. Finally, an infinite set of conservation laws is given
for the resulting hierarchy. 相似文献
3.
The Neumann Type Systems and Algebro-Geometric Solutions of a System of Coupled Integrable Equations
A system of (1+1)-dimensional coupled integrable equations is decomposed into a pair of new Neumann type systems that separate
the spatial and temporal variables for this system over a symplectic submanifold. Then, the Neumann type flows associated
with the coupled integrable equations are integrated on the complex tour of a Riemann surface. Finally, the algebro-geometric
solutions expressed by Riemann theta functions of the system of coupled integrable equations are obtained by means of the
Jacobi inversion. 相似文献
4.
A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations. 相似文献
5.
Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable
symplectic map and finite-dimensional integrable systems are given
by nonlinearization method. The binary Bargmann constraint gives
rise to a Bäcklund transformation for the resulting
integrable lattice equations. At last, conservation laws of the
hierarchy are presented. 相似文献
6.
Comment on “A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map“
Comment on a recent paper on Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 523-528. 相似文献
7.
《Journal of Nonlinear Mathematical Physics》2013,20(1-2):221-230
Abstract New completely integrable generalized C. Neumann systems on several symplectic submanifolds are presented, and the relations between the generalized C. Neumann systems and the spectral problems are further discussed in this paper. In particular, a new eigenvalue problem is proposed in Part 3.3. 相似文献
8.
A semi-direct sum of two Lie algebras of four-by-four
matrices is presented, and a discrete four-by-four matrix spectral problem
is introduced. A hierarchy of discrete integrable coupling systems
is derived. The obtained integrable coupling systems are all written in
their Hamiltonian forms by the discrete variational identity. Finally, we
prove that the lattice equations in the obtained integrable coupling systems
are all Liouville integrable discrete Hamiltonian systems. 相似文献
9.
XU Xi-Xiang YANG Hong-Xiang LU Rong-Wu 《理论物理通讯》2008,50(12):1269-1275
A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems. 相似文献
10.
CHEN Lan-Xin SUN Ye-Peng ZHANG Jun-Xian 《理论物理通讯》2008,49(3):540-544
A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established. 相似文献
11.
In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives:
The first two are the positive members of the hierarchy, and the first equation was a reduction of an integrable (2+1)-dimensional system (see B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A
102 (1984), 15–17). The third one is the first negative member. All nonlinear equations in the hierarchy are shown to have 3×3 Lax pairs through solving a key 3×3 matrix equation, and therefore they are integrable. Under a constraint between the potential function and eigenfunctions, the 3×3 Lax pair and its adjoint representation are nonlinearized to be two Liouville-integrable Hamiltonian systems. On the basis of the integrability of 6N-dimensional systems we give the parametric solution of all positive members in the hierarchy. In particular, we obtain the parametric solution of the equation u
t
=5
x
u
–2/3. Finally, we present the traveling wave solutions (TWSs) of the above three representative equations. The TWSs of the first two equations have singularities, but the TWS of the 3rd one is continuous. The parametric solution of the 5th-order equation u
t
=5
x
u
–2/3 can not contain its singular TWS. We also analyse Gaussian initial solutions for the equations u
t
=5
x
u
–2/3, and u
xxt
+3u
xx
u
x
+u
xxx
u=0. Both of them are stable. 相似文献
12.
Eleven different types of “maximally superintegrable” Hamiltonian systems on the real hyperboloid (s0)2 – (s1)2 + (s2)2 – (s3)2 = 1 are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space SU(2, 2)/U(2, 1), but to reductions by different maximal abelian subgroups of SU(2, 2). Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the O(2, 2) space. 相似文献
13.
Philippe Di Francesco 《Letters in Mathematical Physics》2011,96(1-3):299-324
In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the Q- and T -systems based on A r . The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings. 相似文献
14.
LOU Sen-Yue 《理论物理通讯》2011,55(5):743-746
After introducing dark parameters into the traditional physical models, some types of new phenomena may be found. An important difficult problem is how to directly observe this kind of physical phenomena. An alternative treatment is to introduce equivalent multiple partner fields. If use this ideal to integrable systems, one may obtain infinitely many new coupled integrable systemsconstituted by the original usual field and partner fields. The idea is illustrated via the celebrate KdV equation. From the procedure, some byproducts can be obtained: A new method to find exact solutions of some types of coupled nonlinear physical problems, say, the perturbation KdV systems, is provided; Some new localized modes such as the staggered modes can be found and some new interaction phenomena like the ghost interaction are discovered. 相似文献
15.
《理论物理通讯》2017,(10)
We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov–Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the(2+1)-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of(2+1)-dimensional expanding dynamical model of the(2+1)-dimensional dynamical system is generated as well. 相似文献
16.
Braid group representations are found in ferrnion systems in one space dimension. Explicit baxterization is performed to find corresponding new trigonometric solu tions of Yang-Baxter equation. The quantum algebra structures implied in the new solutions are discovered. Algebraic Bethe ansatz method is applied to solving these systems. The relationships between these fermion systems and polaron model, spin-1/2 Heisenberg spin chain are discussed. 相似文献
17.
18.
19.
In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given. 相似文献