Conservation laws of some lattice equations
Received date: 11 Dec 2012
Accepted date: 03 Apr 2013
Published date: 01 Oct 2013
Copyright
We derive infinitely many conservation laws for some multidimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schr¨odinger equation, modified lattice Boussinesq equation, Hietarinta’s Boussinesq-type equations, Schwarzian lattice Boussinesq equation, and Toda-modified lattice Boussinesq equation.
Junwei CHENG , Dajun ZHANG . Conservation laws of some lattice equations[J]. Frontiers of Mathematics in China, 0 , 8(5) : 1001 -1016 . DOI: 10.1007/s11464-013-0304-z
1 |
Adler V E, Bobenko A I, Suris Yu B. Classification of integrable equations on quadgraphs. The consistency approach. Comm Math Phys, 2003, 233: 513-543
|
2 |
Atkinson J, Hietarinta J, Nijhoff F W. Seed and soliton solutions for Adler’s lattice equation. J Phys A, 2007, 40: F1-F8
|
3 |
Atkinson J, Hietarinta J, Nijhoff F W. Soliton solutions for Q3. J Phys A, 2008, 41: 142001
|
4 |
Bobenko A I, Suris Yu B. Integrable systems on quad-graphs. Int Math Res Not, 2002, 11: 573-611
|
5 |
Bridgman T, Hereman W, QuispelG R W, van der Kamp P H. Symbolic computation of Lax pairs of partial difference equations using consistency around the cube. Found Comput Math, doi: 10.1007/s10208-012-9133-9
|
6 |
Hietarinta J. Searching for CAC-maps. J Nonlinear Math Phys, 2005, 12(Suppl.2): 223-230
|
7 |
Hietarinta J. Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J Phys A, 2011, 44: 165204
|
8 |
Hietarinta J, Zhang D J. Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization. J Phys A, 2009, 42: 404006
|
9 |
Hietarinta J, Zhang D J. Multisoliton solutions to the lattice Boussinesq equation. J Math Phys, 2010, 51: 033505
|
10 |
Hietarinta J, Zhang D J. Soliton taxonomy for a modification of the lattice Boussinesq equation. SIGMA, 2011, 7: 061
|
11 |
Hydon P E. Conservation laws of partial difference equations with two independent variables. J Phys A, 2001, 34: 10347-10355
|
12 |
Mikhailov A V. From automorphic Lie algebras to discrete integrable systems. In: Programme on Discrete Integrable Systems. Isaac Newton Institute for Mathematical Sciences, Cambridge. 2009, http://www.newton.ac.uk/programmes/DIS/seminars /061714001.html
|
13 |
Mikhailov A V, Wang J P, Xenitidis P. Recursion operators, conservation laws and integrability conditions for difference equations. Theoret Math Phys, 2011, 167: 421-443
|
14 |
Nijhoff F W. On some “Schwarzian” equations and their discrete analogues. In: Fokas A S, Gel’fand I M, eds. Algebraic Aspects of Integrable Systems, In Memory of Irene Dorfman. New York: Birkhäuser, 1996, 237-260
|
15 |
Nijhoff F W, Atkinson J, Hietarinta J. Soliton solutions for ABS lattice equations: I. Cauchy matrix approach. J Phys A, 2009, 42: 404005
|
16 |
Nijhoff F W, Papageorgiou V G, Capel H W, Quispel G R W. The lattice Gel’fand- Dikii hierarchy. Inverse Problems, 1992, 8: 597-621
|
17 |
Nijhoff F W, Quispel G R W, Capel H W. Direct linearization of nonlinear differencedifference equations. Phys Lett A, 1983, 97: 125-128
|
18 |
Nijhoff F W, Walker A J. The discrete and continuous Painlev′e VI hierarchy and the Garnier systems. Glasg Math J, 2001, 43A: 109-123
|
19 |
Rasin A G. Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method. J Phys A, 2010, 43: 235201
|
20 |
Rasin O G, Hydon P E. Conservation laws for NQC-type difference equations. J Phys A, 2006, 39: 14055-14066
|
21 |
Rasin A G, Schiff J. Infinitely many conservation laws for the discrete KdV equation. J Phys A: Math Theor, 2009, 42: 175205
|
22 |
Tongas A, Nijhoff F W. The Boussinesq integrable system: compatible lattice and continuum structures. Glasg Math J, 2005, 47: 205-219
|
23 |
Wadati M, Sanuki H, Konno K. Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Progr Theoret Phys, 1975, 53: 419-436
|
24 |
Walker A. Similarity reductions and integrable lattice equations. DissertationTip, Univ of Leeds, UK, 2001
|
25 |
Xenitidis P. Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type. J Phys A, 2011, 44: 435201
|
26 |
Xenitidis P, Nijhoff F W. Symmetries and conservation laws of lattice Boussinesq equations. Phys Lett A, 2012, 376: 2394-2401
|
27 |
Zhang D J. Conservation laws of the two-dimensional Toda lattice hierarchy. J Phys Soc Jpn, 2002, 71: 2583-2586
|
28 |
Zhang D J, Chen D Y. The conservation laws of some discrete soliton systems. Chaos Solitons Fractals, 2002, 14: 573-579
|
29 |
Zhang D J, Cheng J W, Sun Y Y. Deriving conservation laws for ABS lattice equations from Lax pairs. arXiv: 1210.3454v1
|
30 |
Zhu Z N, Wu X N, Xue W M, Zhu Z M. Infinitely many conservation laws for the Blaszak-Marciniak four-field integrable lattice hierarchy. Phys Lett A, 2002, 296: 280-288
|
/
〈 | 〉 |