Research Article
Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs
Fajun yu,
Shuo Feng,
Corresponding Author
Fajun yu
School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034 China
Correspondence to: Fajun yu, School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China.
E-mail: [email protected]
Search for more papers by this authorShuo Feng
School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034 China
Search for more papers by this authorFajun yu,
Shuo Feng,
Corresponding Author
Fajun yu
School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034 China
Correspondence to: Fajun yu, School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China.
E-mail: [email protected]
Search for more papers by this authorShuo Feng
School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034 China
Search for more papers by this authorAbstract
The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.
References
- 1Akhmediev N, Ankiewicz A. Solitons: Nonlinear Pulses and Beams. Chapman and Hall: London, 1997.
- 2Kivshar YS, Agrawal P. Optical Solitons: From Fibers to Photonic Crystals. Academic Press: New York, 2003.
- 3Barnett MP, Capitani JF, Von Zur Gathen J, Gerhard J. Symbolic calculation in chemistry: selected examples. International Journal of Quantum Chemistry 2004; 100: 80–104.
- 4Matveev VB, Salle MA. Darboux Transformation and Solitons. Springer: Berlin, 1991.
10.1007/978-3-662-00922-2 Google Scholar
- 5Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge Univ Press: New York, 1991.
10.1017/CBO9780511623998 Google Scholar
- 6Wadati M. Wave propagation in nonlinear lattice. I. Journal of the Physical Society of Japan 1975; 38: 673–680.
- 7Gao YT, Tian B. Reply to: “Comment on: ‘Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation’ ”. Physics Letters A 2007; 361: 520.
- 8Weiss J, Tabor M, Carnevale G. The Painleve property for partial differential equations. Journal of Mathematical Physics 1983; 24: 522–526.
- 9Hirota R. The Direct Method in Soliton Theory. Cambridge Univ Press: Cambridge, 2004.
10.1017/CBO9780511543043 Google Scholar
- 10Deift P, Trubowitz E. Inverse scattering on the line. Communications on Pure and Applied Mathematics 1979; 32: 121–251.
- 11Matveev VB, Salle MA. Darboux Transformations and Solitons. Springer-Verlag: Berlin, 1991.
10.1007/978-3-662-00922-2 Google Scholar
- 12Gu CH, Hu HS, Zhou Z. Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Springer: London, 2006.
- 13Terng CL, Uhlenbeck K. Backlund transformations and loop group actions. Communications on Pure and Applied Mathematics 2000; 53: 1–75.
- 14Novikov SP, Manakov SV, Zakharov VE, Pitaevskii LP. Theory of Solitons: The Inverse Scattering Method. Springer: Verlag Berlin Heidelberg, 1984.
- 15Gu CH, Hu HS, Zhou ZX. Darboux Transform in Soliton Theory and Its Geometric Applications. Shanghai Scientific Technical Publishers: Shanghai, 1999.
- 16Ding HY, Xu XX, Zhao XD. A hierarchy of lattice soliton equations and its Darboux transformation. Chinese Physics 2004; 3: 125.
- 17Wu YT, Geng XG. A new hierarchy integrable differential–difference equations and Darboux transformation. Journal of Physics A: Mathematical and General 1998; 31: L677–L684.
- 18Xu XX, Yang HX, Sun YP. Darboux transformation of the modiffed Toda lattice equation. Modern Physics Letters B 2006; 20: 641–648.
- 19Xu XX. An integrable coupling family of Merola–Ragnisco–Tu lattice systems, its Hamiltonian structure and related nonisospectral integrable lattice family. Physics Letters A 2010; 374: 401–410.
- 20Xu XX. Solving an integrable coupling system of Merola–Ragnisco–Tu lattice equation by Darboux transformation of Lax pair. Communications in Nonlinear Science 2015; 23: 192–201.
- 21Tu GZ. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. Journal of Mathematical Physics 1989; 30: 330–338.
- 22Tu GZ. A trace identity and its applications to the theory of discrete integrable systems. Journal of Physics A: Mathematical and General 1990; 23: 3903–3922.
- 23Tu GZ, Ma WX. An algebraic approach for extending Hamiltonian operators. Journal of Partial Differential Equations 1992; 3: 53–56.
- 24Ablowitz MJ, Ladik JF. Nonlinear differential–difference equations. Journal of Mathematical Physics 1975; 16: 598–603.
- 25Geng XG, Dai HH, Zhu JY, Wang HY. Decomposition of the discrete Ablowitz–Ladik hierarchy. Studies in Applied Mathematics 2007; 118: 281–312.
- 26Yu FJ. Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz–Ladik equation. Physical Review E 2015; 91: 032914 (8 pages).
- 27Bruschi M, Ragnisco O. Recurdion operator and Backlund-transformations for the Ruijsenaars–Toda lattice. Physics Letters A 1988; 129: 21–25.
- 28Bruschi M, Ragnisco O. Direct and inverse spectral problem for the infinite Ruijsenaars–Toda lattice. Inverse Problems 1989; 5: 389–405.
- 29Suris YB. New integrable systems related to the relativistic Toda lattice. Journal of Physics A: Mathematical and General 1997; 30: 1745–1761.
- 30Wadati M. Transformation theories for nonlinear discrete systems. Progress of Supplement Theoretical Physics 1976; 59: 36–63.
10.1143/PTPS.59.36 Google Scholar
- 31Damianou PA. The Voltera model and its relation to the Toda lattice. Physics Letters A 1991; 155: 126–132.
- 32Damianou PA, Fernandes RL. From the Toda lattice to the Volterra lattice and back. Reports on Mathematical Physics 2002; 50: 361–378.
- 33Tsuchidat T, Wadati M. Bi-Hamiltonian structure of modified Volterra model. Chaos Solitons Fractals 1998; 9: 869–873.
- 34Yan ZY. Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm. Nonlinear Analysis 2006; 64: 1798–1811.
- 35Hu XB, Zhu ZN. Some new results on the Blaszak–Marciniak lattice: Backlund transformation and nonlinear superposition formula. Journal of Mathematical Physics 1998; 39: 4766–4772.
- 36Zhu ZN, Zhu ZM, Wu XN, Xue WM. New matrix Lax representation for a Blaszak–Marciniak four-field lattice hierarchy and its infinitely many conservation laws. Physical Society of Japan 2002; 71: 1864–1869.
- 37Ma WX, Fuchssteiner B. Integrable theory of the perturbation equations. Chaos, Solitons and Fractals 1996; 7: 1227–1250.
- 38Ma WX, Zhou RG. Nonlinearization of spectral problems of the perturbation KdV systems. Physica A 2001; 296: 60–74.
- 39Ma WX. A bi-Hamiltonian formulation for triangular systems by perturbations. Journal of Mathematical Physics 2002; 43: 1408–1421.
- 40Fan EG, Zhang Y. A simple method for generating integrable hierarchies with multi-potential functions. Chaos Solitons Fractals 2005; 25: 425–439.
- 41Ma WX, Fuchssteiner B. The bi-Hamiltonian structures of the perturbation equations of KdV hierarchy. Physics Letters A 1996; 213: 49–55.
- 42Ma WX. Enlarging spectral problems to construct integrable couplings of soliton equations. Physics Letters A 2003; 316: 72–76.
- 43Ma WX. Integrable couplings of vector AKNS soliton equations. Journal of Mathematical Physics 2005; 46: 033507 (19 pages).
- 44Guo FK, Zhang YF. A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling. Journal of Mathematical Physics 2003; 44: 5793–5803.
- 45Zhang YF. A generalized multi-component Glachette–Johnson (GJ) hierarchy and its integrable coupling system. Chaos Solitons and Fractals 2004; 21: 305–310.
- 46Ma WX, Xu XX, Zhang YF. Semidirect sums of Lie algebras and discrete integrable couplings. Journal of Mathematical Physics 2006; 47: 053501 (16 pages).
- 47Ma WX, Chen M. Hamiltonian and quasi-Hamiltonian structures associated with semidirect sums of Lie algebras. Journal of Physics A: Mathematical and General 2006; 39: 10787–10801.
- 48Tian SF, Zhang HQ. Soliton solutions by Darboux transformation and some reductions for a new Hamiltonian lattice hierarchy. Physica Scripta 2010; 82: 015008 (8 pages).
