A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy

DOI: 10.4236/jamp.2015.310157   PDF   HTML   XML   2,428 Downloads   2,679 Views  


A new lattice hierarchy related to Ragnisco-Tu equation is proposed and its gauge equivalence to Ragnisco-Tu equation is proven. As an application of gauge transformation, we construct Darboux transformation (DT) of this new equation through DT of Ragnisco-Tu equation. An explicit exact solution is presented as an example.

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Liu, Y. , Hu, C. and Dai, J. (2015) A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy. Journal of Applied Mathematics and Physics, 3, 1282-1294. doi: 10.4236/jamp.2015.310157.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Chen, D.Y. (2006) Introduction of Soliton. Science Press, Beijing. (In Chinese)
[2] Chen, D.Y. and Zeng, Y.B. (1985) The Trans-formation Operator of Nonlinear Evolution Equations I. Chinese Annals of Mathematics, 6B, 71-82.
[3] Chen, D.Y. and Zeng, Y.B. (1985) The Transformation Operator of Nonlinear Evolution Equations III. Acta Mathematica, 28, 161-173. (In Chinese)
[4] Chen, D.Y. and Li, Y.S. (1986) The Transformation Operator of Nonlinear Evolution Equations IV. Acta Mathematica, 29, 127-134. (In Chinese)
[5] Chen, D.Y. and Li, Y.S. (1986) The Transformation Operator of Nonlinear Evolution Equations V. Acta Mathematica Sinica, New Series, 2, 343-356.
[6] Chau, L.L., Shaw, J.C. and Yen, H.C. (1992) Solving the KP Hierarchy by Gauge Transformations. Communications in Theoretical Physics, 149, 263-278.
[7] Zhang, H.W., Tu, G.Z., et al. (1991) Symmetries, Conserved Quantities, and Hier-archies for Some Lattice Systems with Soliton Structure. Journal of Mathematical Physics, 32, 1908-1918.
[8] Zeng, Y.B. (1998) Restried Flows of a Hierarchy of Integrable Discrete Systems. Acta Mathematicae Applicatae Sinica, 14, 176-184.
[9] Liu, Y.Q. and Chen, D.Y. (2011) The Exact Solutions to a Ragnisco-Tu Hierarchy with Self-Consistent Sources. Nonlinear Analysis, 74, 5223-5237.
[10] Xu, X.X. (2010) An Integrable Coupling Family of Merola-Ragnisco-Tu Lattice Systems, Its Hamiltonian Structure and Related Nonisospectral Integrable Lattice Family. Physics Letters A, 374, 401-410.
[11] Zeng, X. and Geng, X.G. (2014) Algebro-Geometric Solutions of the Discrete Ragnisco-Tu Hierarchy. Reports on Mathematical Physics, 73, 17-48.
[12] Xu, X.X. (2015) Solving an Integrable Coupling System of Mer-ola-Ragnisco-Tu Lattice Equation by Darboux Transformation of Lax Pair. Communications in Nonlinear Science and Numerical Simulation, 23, 192-201.
[13] Xu, X.X., Yang, H.X. and Sun, Y.P. (2006) Dar-boux Transformation of the Modified Toda Lattice Equation. Modern Physics Letters B, 20, 641-648.
[14] Hon, Y.C. and Fan, E.G. (2009) Quasi-Periodic Solutions for Modified Toda Lattice Equation. Chaos, Solitons and Fractals, 40, 1297-1308.
[15] Spiridonov, V. and Zhedanov, A. (1995) Discrete Darboux Transformations, the Discrete-Time Toda Lattice, and the Askey-Wilson Polynomials. Methods and Applications of Analysis, 2, 369-398.
[16] Fan, E.G. (2001) Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrödinger Equations. Com-munications in Theoretical Physics, 35, 651-656.
[17] Zeng, Y.B., Shao, Y.J. and Ma, W.X. (2002) Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources. Communications in Theoretical Physics, 38, 641-648.
[18] He, J.S., Zhang, L., Chen, Y. and Li, Y.S. (2006) Determinant Representation of Darboux Transformation for the AKNS System. Science in China Series A: Mathematics, 49, 1867-1878.
[19] Zhou, Z.X. (2007) Darboux Transformations for Some Two Dimensional Affine Toda Equations. The International Congress of Chinese Mathematicians (ICCM), III, 405-416.
[20] Wang, L., Gao, Y.T., Gai, X.L., Yu, X. and Sun, Z.Y. (2010) Vadermonde-Type Odd-Soliton Solutions for the Whitham-Broer-Kaup Model in the Shallow Water Small-Amplitude Regime. Journal of Nonlinear Mathematical Physics, 17, 197-211.
[21] Geng, X.G. and He, G.L. (2010) Darboux Transformation and Explicit Solutions for the Satsuma-Hirota Coupled Equation. Applied Mathematics and Computation, 216, 2628-2634.
[22] Xu, G.H. (2014) N-Fold Darboux Transformation of the Jaulent-Miodek Equation. Applied Mathematics, 5, 2657-2663.


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