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1.
ZHANG Ya-Xing ZHANG Hai-Qiang LI Juan XU Tao ZHANG Chun-Yi TIAN Bo 《理论物理通讯》2008,49(4):833-838
In this paper, we put our focus on a variable-coe~cient fifth-order Korteweg-de Vries (fKdV) equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out. 相似文献
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3.
Using the relation between the mKdV equation and the KdV-mKdV equation, we derive non-singular rational solutions for the mKdV equation. The solutions are given in terms of Wronskians. Dynamics of some solutions is investigated by means of asymptotic analysis. Wave trajectories of high order rational solutions are asymptotically governed by cubic curves. 相似文献
4.
CAI Ke-Jie TIAN Bo ZHANG Cheng ZHANG Huan MENG Xiang-Hua LU Xing GENG Tao LIU Wen-Jun 《理论物理通讯》2008,50(11):1185-1188
By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the (N - 1)- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique. 相似文献
5.
The modified Korteweg-de Vries (mKdV) typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics. 相似文献
6.
In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is investigated. With the help of symbolic computation, the N-soliton solution is derived through the Hirota method. Then the bilinear Bäcklund transformations and Lax pairs are presented. At last, we show some interactions of solitary waves. 相似文献
7.
CAI Ke-Jie TIAN Bo ZHANG Cheng ZHANG Huan MENG Xiang-Hua LÜ Xing GENG Tao LIU Wen-Jun 《理论物理通讯》2008,50(5):1185-1188
By the symbolic computation and Hirota method, the bilinear form and
an auto-Bäcklund transformation for a variable-coefficient
Korteweg-de Vries equation with nonuniformities are given. Then, the
N-solitonic solution in terms of Wronskian form is obtained and
verified. In addition, it is shown that the (N-1)- and N-solitonic
solutions satisfy the
auto-Bäcklund transformation through the Wronskian technique. 相似文献
8.
In this article, we study the Lax pairs of (2+1)-dimensional equation: the
modified generalized dispersive long wave (MGDLW) equation. Based on the
well-known binary Darboux transformation, we dig out the recursion formulas of the
first part of the Lax pairs. Then by further discussion and doing some revisional work,
we make the recursion formulas fit for the second part of Lax pairs. At last, some
solutions to the MGDLW equation are worked out by using the recursion formula. 相似文献
9.
BAI ChengLin 《理论物理通讯》2002,37(6):645-648
Using the extended homogeneous balance method, which is very concise and primary, Lax pairs and Backlund transformation for most nonlinear evolution equations, such as the compound KdV-Burgers equation and nonlinear diffusion equation are obtained. 相似文献
10.
Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries (ABmKdV) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-mKdV equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, "the shifted parity and delayed time reversal to describe two-place events". 相似文献
11.
Extension of the Painlevé equations to noncommutative spaces has been extensively investigated in the theory of integrable systems. An interesting topic is the exploration of some remarkable aspects of these equations, such as the Painlevé property, the Lax representation and the Darboux transformation, and their connection to well-known integrable equations. This paper addresses the Lax formulation, the Darboux transformation and a quasideterminant solution of the noncommutative form of Painlevé’s second equation introduced by Retakh and Rubtsov [V. Retakh, V. Rubtsov, Noncommutative Toda chain, Hankel quasideterminants and Painlevé II equation, J. Phys. A Math. 43 (2010) 505204]. 相似文献
12.
Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie's first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th B?cklund transformation in the form of the determinant. Moreover, the P_sT_d(the shifted parity and delayed time reversal) symmetric exact solutions(including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation. 相似文献
13.
The dynamical behaviour of the generalized Korteweg-de
Vries (KdV) equation under a periodic perturbation is investigated
numerically. The bifurcation and chaos in the system are observed by
applying bifurcation diagrams, phase portraits and Poincaré maps.
To characterise the chaotic behaviour of this system, the spectra of
the Lyapunov exponent and Lyapunov dimension of the attractor are also
employed. 相似文献
14.
With the aid of a gauge transformation, we propose a Darboux transformation for a four-component KdV equation. As an application, we obtain some explicit solutions for the four-component KdV equation. 相似文献
15.
The prolongation structure methodologies of Wahlquist--Estabrook [Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1] for nonlinear differential equations are applied to a variable-coefficient KdV equation. Based on the obtained prolongation structure, a Lie algebra with five parameters is constructed. Under certain conditions, a Lie algebra representation and three kinds of Lax pairs for the variable- coefficient KdV equation are derived. 相似文献
16.
JU Guo-Xing 《理论物理通讯》2003,39(3):285-290
Using the colored Yang-Baxter equation, the Lax pairs for the system with color parameters and periodic boundary conditions are derived. The explicit expressions of the Lax pairs for the six-vertex standard R matrices of both the Baxter and free-Fermion types are given. 相似文献
17.
A base-equation method is implemented to realize the hereditary algebra of the Korteweg-de Vries (KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of our approach is that, it can in a rather natural way, predict other nonlinear
evolution equations which admit local conservation laws. Significantly enough, base functions themselves are found to provide
a basis to regard the KdV-like equations as higher order degenerate bi-Lagrangian systems. 相似文献
18.
XUAN Qi-Fei ZHANG Da-Jun 《理论物理通讯》2008,50(7):13-16
In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via the Backlund transformation (BT) and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN (t) for the original diagonal one. 相似文献
19.
WU Hong-Xia ZENG Yun-Bo FAN Tian-You 《理论物理通讯》2008,49(3):529-534
Darboux transformation (DT) provides us with a comprehensive approach to construct the exact and explicit solutions to the negative extended KdV (eKdV) equation, by which some new solutions such as singular soliton, negaton, and positon solutions are computed for the eKdV equation. We rediscover the soliton solution with finiteamplitude in [A.V. Slyunyaev and E.N. Pelinovskii, J. Exp. Theor. Phys. 89 (1999) 173] and discuss the difference between this soliton and the singular soliton. We clarify the relationship between the exact solutions of the eKdV equation and the spectral parameter. Moreover, the interactions of singular two solitons, positon and negaton, positon and soliton, and two positons are studied in detail. 相似文献
20.
XU Chang-Zhi 《理论物理通讯》2006,46(9)
Variable separation approach is introduced to solve the (2 1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model (24). Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately. 相似文献