共查询到20条相似文献,搜索用时 31 毫秒
1.
We expose a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. 相似文献
2.
On Classification of Integrable Nonevolutionary Equations 总被引:1,自引:0,他引:1
Alexander V. Mikhailov Vladimir S. Novikov Jing Ping Wang 《Studies in Applied Mathematics》2007,118(4):419-457
We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the homogeneous integrable equations of fourth and sixth order (in the space derivative) equations, as well as we have found three new tenth-order integrable equations. To prove the integrability we provide the corresponding bi-Hamiltonian structures and recursion operators. 相似文献
3.
Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system. 相似文献
4.
A. V. Mikhailov Jing Ping Wang P. Xenitidis 《Theoretical and Mathematical Physics》2011,167(1):421-443
We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion
operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation
laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability
conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris
equations. 相似文献
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6.
We show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential-delay equations. All members of these hierarchies of lattice and differential-delay equations commute. It is thus seen that differential-delay hierarchies provide a broader context within which to place lattice hierarchies. 相似文献
7.
V. S. Gerdjikov G. G. Grahovski A. V. Mikhailov T. I. Valchev 《Theoretical and Mathematical Physics》2011,167(3):740-750
We analyze and compare methods for constructing the recursion operators for a special class of integrable nonlinear differential
equations related to symmetric spaces of the type A.III in Cartan’s classification and having additional reductions. 相似文献
8.
Abdul‐Majid Wazwaz 《Mathematical Methods in the Applied Sciences》2019,42(5):1553-1560
In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation. 相似文献
9.
10.
R. Sahadevan S. Khousalya L. Nalini Devi 《Journal of Mathematical Analysis and Applications》2005,308(2):636-655
A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation. 相似文献
11.
We provide examples to extend a recent conjecture concerning the relation between zero curvature representations and nonlocal
terms of inverse recursion operators to all recursion operators in dimension two. Namely, we conjecture that nonlocal terms
of recursion operators are always related to a suitable zero-curvature representation, not necessarily depending on a parameter
or taking values in a semisimple algebra. In particular, the conventional pseudodifferential recursion operators correspond
to abelian Lie algebras.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 23–33,
2006. 相似文献
12.
B. G. Konopelchenko 《Acta Appl Math》1989,16(1):75-116
The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered. 相似文献
13.
A class of nonlocal symmetries of the Camassa-Holm type equations
with bi-Hamiltonian structures, including the Camassa-Holm equation,
the modified Camassa-Holm equation, Novikov equation and
Degasperis-Procesi equation, is studied. The nonlocal symmetries are
derived by looking for the kernels of the recursion operators and
their inverse operators of these equations. To find the kernels of
the recursion operators, the authors adapt the known
factorization results for the recursion operators of the KdV,
modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the
explicit Liouville correspondences between the KdV and Camassa-Holm
hierarchies, the modified KdV and modified Camassa-Holm hierarchies,
the Novikov and Sawada-Kotera hierarchies, as well as the
Degasperis-Procesi and Kaup-Kupershmidt hierarchies. 相似文献
14.
《Mathematical and Computer Modelling》1997,25(8-9):91-100
We present algorithms and describe CA-packages to compute the infinitesimal generators of infinite-dimensional symmetry groups for integrable PDEs (evolution equations) in one space and one time dimension. Here, integrable is meant in the sense that the vector field defining the equation is a member of the abelian part of some infinite-dimensional Virasoro algebra. The method of computation is completely different from the usual prolongation method, no determining equations are solved. Instead, all necessary generators of the finitely generated Virasoro algebra are computed from one given element by direct Lie algebra methods. The implementation of the algorithms in MuPAD is described. A sample session is included in which the recursion structures of the KdV and the Krichever-Novikov equations are computed. 相似文献
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16.
We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1?1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 ?λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1\(\tilde U\) = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution \(\tilde U\) of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations. 相似文献
17.
Multi‐component integrable couplings for the Ablowitz–Kaup–Newell–Segur and Volterra hierarchies
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Shoufeng Shen Chunxia Li Yongyang Jin Shuimeng Yu 《Mathematical Methods in the Applied Sciences》2015,38(17):4345-4356
A kind of N × N non‐semisimple Lie algebra consisting of triangular block matrices is used to generate multi‐component integrable couplings of soliton hierarchies from zero curvature equations. Two illustrative examples are made for the continuous Ablowitz–Kaup–Newell–Segur hierarchy and the semi‐discrete Volterra hierarchy, together with recursion operators. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
18.
In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h. 相似文献
19.
A variety of (3 + 1)‐dimensional Burgers equations derived by using the Burgers recursion operator
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Abdul‐Majid Wazwaz 《Mathematical Methods in the Applied Sciences》2015,38(12):2642-2649
A new variety of (3 + 1)‐dimensional Burgers equations is presented. The recursion operator of the Burgers equation is employed to establish these higher‐dimensional integrable models. A generalized dispersion relation and a generalized form for the one kink solutions is developed. The new equations generate distinct solitons structures and distinct dispersion relations as well. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
20.
A. M. Verbovetsky R. Vitolo P. Kersten I. S. Krasil’shchik 《Theoretical and Mathematical Physics》2012,171(2):600-615
We consider a third-order generalized Monge-Ampère equation uyyy ? u xxy 2 + uxxxuxyy = 0, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws. 相似文献