Conservation laws of some lattice equations

Abstract

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 Schrödinger equation, modified lattice Boussinesq equation, Hietarinta’s Boussinesq-type equations, Schwarzian lattice Boussinesq equation, and Toda-modified lattice Boussinesq equation.

Keywords

Conservation law / Lax pair / multi-dimensionally consistent lattice equation / discrete integrable system

References

[1] Adler V E, Bobenko A I, Suris Y 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 Y B. Integrable systems on quad-graphs. Int Math Res Not, 2002, 11: 573-611

[5] Bridgman T, Hereman W, Quispel G 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. Programme on Discrete Integrable Systems, 2009, Cambridge: Isaac Newton Institute for Mathematical Sciences

[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. Fokas A S, Gel’fand I M. On some “Schwarzian” equations and their discrete analogues. Algebraic Aspects of Integrable Systems, In Memory of Irene Dorfman, 1996, New York: Birkhäuser 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 Painlevé 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, 2001, UK: Univ of Leeds

[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