Frontiers of Mathematics in China >
Spectrum transformation and conservation laws of lattice potential KdV equation
Received date: 27 Dec 2015
Accepted date: 22 Apr 2016
Published date: 20 Feb 2017
Copyright
Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.
Senyue LOU , Ying SHI , Da-jun ZHANG . Spectrum transformation and conservation laws of lattice potential KdV equation[J]. Frontiers of Mathematics in China, 2017 , 12(2) : 403 -416 . DOI: 10.1007/s11464-016-0542-y
1 |
Ablowitz M J, Ladik J F. Nonlinear differential-difference equations and Fourier analysis. J Math Phys, 1976, 17: 1011–1018
|
2 |
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
|
3 |
Boiti M, Pempinelli F, Prinari B, Spire A. An integrable discretization of KdV at large times. Inverse Problems, 2001, 17: 515–526
|
4 |
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, 2013, 13: 517–544
|
5 |
Butler S. Inverse Scattering Transform Method for Lattice Equations. Ph D Thesis, University of Sydney, 2012
|
6 |
Butler S. Multidimensional inverse scattering of integrable lattice equations. Nonlinearity, 2012, 25: 1613–1634
|
7 |
Butler S, Joshi N. An inverse scattering transform for the lattice potential KdV equation. Inverse Problems, 2010, 26: 115012
|
8 |
Cheng J W, Zhang D J. Conservation laws of some lattice equations. Front Math China, 2013, 8: 1001–1016
|
9 |
Habibullin I, Yangubaeva M. Formal diagonalization of the discrete Lax operators and construction of conserved densities and symmetries for dynamical systems. Theoret Math Phys, 2013, 177: 1655–1679
|
10 |
Hietarinta J. Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J Phys A, 2011, 44: 165204
|
11 |
Hietarinta J, Zhang D J. Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization. J Phys A, 2009, 42: 404006
|
12 |
Hietarinta J, Zhang D J. Multisoliton solutions to the lattice Boussinesq equation. J Math Phys, 2010, 51: 033505
|
13 |
Hietarinta J, Zhang D J. Soliton taxonomy for a modification of the lattice Boussinesq equation. SIGMA Symmetry Integrability Geom Methods Appl, 2011, 7: 061
|
14 |
Levi D, Petrera M, Scimiterna C, Yamilov R. On Miura transformations and Volterratype equations associated with the Adler-Bobenko-Suris equations. SIGMA Symmetry Integrability Geom Methods Appl, 2008, 4: 077
|
15 |
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
|
16 |
Nijhoff F W. Lax pair for the Adler (lattice Krichever-Novikov) system. Phys Lett A, 2002, 297: 49–58
|
17 |
Nijhoff F W, Atkinson J, Hietarinta J. Soliton solutions for ABS lattice equations. I. Cauchy matrix approach. J Phys A, 2009, 42: 404005
|
18 |
Nijhoff F W, Capel H W. The discrete Korteweg-de Vries equation. Acta Appl Math, 1995, 39: 133–158
|
19 |
Nijhoff F W, Quispel G R W, Capel H W. Direct linearization of nonlinear differencedifference equations. Phys Lett A, 1983, 97: 125–128
|
20 |
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
|
21 |
Novikov S, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons: the Inverse Scattering Method. New York: Consult Bureau, 1984
|
22 |
Rasin A G. Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method. J Phys A, 2010, 43: 235201
|
23 |
Rasin A G, Schiff J. Infinitely many conservation laws for the discrete KdV equation. J Phys A, 2009, 42: 175205
|
24 |
Rasin O G, Hydon P E. Conservation laws of discrete Korteweg-de Vries equation. SIGMA Symmetry Integrability Geom Methods Appl, 2005, 1: 026
|
25 |
Rasin O G, Hydon P E. Conservation laws for integrable difference equations. J Phys A, 2007, 40: 12763–12773
|
26 |
Xenitidis P, Nijhoff F W. Symmetries and conservation laws of lattice Boussinesq equations. Phys Lett A, 2012, 376: 2394–2401
|
27 |
Xenitidis P. Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type. J Phys A, 2011, 44: 435201
|
28 |
Zakharov V, Shabat A. Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Sov Phys JETP, 1972, 34: 62–69
|
29 |
Zhang D J, Cheng J W, Sun Y Y. Deriving conservation laws for ABS lattice equations from Lax pairs. J Phys A, 2013, 46: 265202
|
30 |
Zhang D J, Zhao S L, Nijhoff F W. Direct linearization of an extended lattice BSQ system. Stud Appl Math, 2012, 129: 220–248
|
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