Journal ArticleDOI
From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation
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In this article, the quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev-Petviashvili equation is split into three Hamiltonian systems (H0, H1), H2), while that of the special Toda equation is separated into (H 0),(H 1) plus the discrete flow generated by the symplectic map S. The explicit theta function solutions are obtained by resorting to this separation technique.Abstract:
The nonlinearization of the eigenvalue problems associated with the Toda hierarchy and the coupled Korteweg-de Vries (KdV) hierarchy leads to an integrable symplectic map S and an integrable Hamiltonian system (H0), respectively. It is proved that S and (H0) have the same integrals {Hk}. The quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev-Petviashvili equation is split into three Hamiltonian systems (H0),(H1),(H2), while that of the special (2 + 1)-dimensional Toda equation is separated into (H0),(H1) plus the discrete flow generated by the symplectic map S. A clear evolution picture of various flows is given through the `window' of Abel-Jacobi coordinates. The explicit theta-function solutions are obtained by resorting to this separation technique.read more
Citations
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Decomposition of the (2 + 1)- dimensional Gardner equation and its quasi-periodic solutions
Xianguo Geng,Cewen Cao +1 more
TL;DR: In this article, the (2 + 1)-dimensional Gardner equation is decomposed into the first two non-trivial soliton systems in the hierarchy, and two compatible Hamiltonian systems of ordinary differential equations.
Journal ArticleDOI
Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations
TL;DR: In this article, the (2+1)-dimensional breaking soliton equation, the coupled KP equation with three potentials and a new (3+ 1)-dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the (1+1)dimensional AKNS equations.
Journal ArticleDOI
Decomposition of the Discrete Ablowitz–Ladik Hierarchy
Xianguo Geng,H. H. Dai,Junyi Zhu +2 more
TL;DR: In this paper, the nonlinearization approach of Lax pairs is extended to the discrete Ablowitz-Ladik hierarchy, and a new symplectic map and a class of new finite-dimensional Hamiltonian systems are derived, which are further proved to be integrable in the Liouville sense.
Journal ArticleDOI
Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable
TL;DR: In this paper, the quasiperiodic solution of the 2+1 dimensional Burgers equation with a discrete variable is obtained through three steps: decomposition into a symplectic map plus two finite-dimensional Hamiltonian systems; straightening out of both the discrete and the continuous flows in the Jacobian variety; inversion into the original variables.
Journal ArticleDOI
Variable separation approach for a differential-difference system: special Toda equation
TL;DR: In this paper, a bilinear variable separation approach is used to construct some special solutions for a differential-difference Toda equation, and abundant semi-discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions.
References
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Hill’s Operator and Hyperelliptic Function Theory in the Presence of Infinitely Many Branch Points
H. P. McKean,Eugene Trubowitz +1 more
TL;DR: The class of k-times continuously differentiable real-valued functions of period 1 is defined in this article, where the Hill's operator − d2∕dx2 + q(x, λ) with a fixed q of class (C_{1}^{\infty }) is defined.
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The constraint of the Kadomtsev-Petviashvili equation and its special solutions
TL;DR: By constraining the potential of the Kadomtsev-Petviashvili (KP) equation to its co-invariants expressed in terms of the squared eigenfunctions, the KP equation is reduced to a (1 + 1)-dimensional system consisting of the generalized multicomponent nonlinear Schrodinger and modified Korteweg-de Vries equations as mentioned in this paper.
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Relation between the Kadometsev–Petviashvili equation and the confocal involutive system
TL;DR: In this article, the special quasiperiodic solution of the (2+1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem.
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(1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems
TL;DR: In this paper, the authors consider linear problems associated with integrable systems in 2+1 dimensions and obtain generating functions for symmetries from the bilocal approach. But they do not consider symmetric systems.