RESEARCH ARTICLE

Spectrum transformation and conservation laws of lattice potential KdV equation

  • Senyue LOU 1 ,
  • Ying SHI 2 ,
  • Da-jun ZHANG , 3
Expand
  • 1. Faculty of Science, Ningbo University, Ningbo 315211, China
  • 2. School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
  • 3. Department of Mathematics, Shanghai University, Shanghai 200444, China

Received date: 27 Dec 2015

Accepted date: 22 Apr 2016

Published date: 20 Feb 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

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.

Cite this article

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

DOI

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

DOI

3
Boiti M, Pempinelli F, Prinari B, Spire A. An integrable discretization of KdV at large times. Inverse Problems, 2001, 17: 515–526

DOI

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

DOI

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

DOI

7
Butler S, Joshi N. An inverse scattering transform for the lattice potential KdV equation. Inverse Problems, 2010, 26: 115012

DOI

8
Cheng J W, Zhang D J. Conservation laws of some lattice equations. Front Math China, 2013, 8: 1001–1016

DOI

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

DOI

10
Hietarinta J. Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J Phys A, 2011, 44: 165204

DOI

11
Hietarinta J, Zhang D J. Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization. J Phys A, 2009, 42: 404006

DOI

12
Hietarinta J, Zhang D J. Multisoliton solutions to the lattice Boussinesq equation. J Math Phys, 2010, 51: 033505

DOI

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

DOI

16
Nijhoff F W. Lax pair for the Adler (lattice Krichever-Novikov) system. Phys Lett A, 2002, 297: 49–58

DOI

17
Nijhoff F W, Atkinson J, Hietarinta J. Soliton solutions for ABS lattice equations. I. Cauchy matrix approach. J Phys A, 2009, 42: 404005

DOI

18
Nijhoff F W, Capel H W. The discrete Korteweg-de Vries equation. Acta Appl Math, 1995, 39: 133–158

DOI

19
Nijhoff F W, Quispel G R W, Capel H W. Direct linearization of nonlinear differencedifference equations. Phys Lett A, 1983, 97: 125–128

DOI

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

DOI

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

DOI

23
Rasin A G, Schiff J. Infinitely many conservation laws for the discrete KdV equation. J Phys A, 2009, 42: 175205

DOI

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

DOI

26
Xenitidis P, Nijhoff F W. Symmetries and conservation laws of lattice Boussinesq equations. Phys Lett A, 2012, 376: 2394–2401

DOI

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

DOI

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

DOI

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

DOI

Outlines

/