Regular solution and lattice Miura transformation of bigraded Toda hierarchy

Abstract

The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy (BTH). As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the (1, 2)-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the (1, 2)-BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of (N, 1)-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of (N, 1)-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation (a useful ecological competition model) are given.

Keywords

Regular solution / Lattice Miura transformation / Bigraded Toda hierarchy / Moment polytope / Volterra lattice

References

[1] Toda M. Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan, 1967, 22: 431-436

[2] Toda M. Nonlinear waves and solitons, 1989, Holland: Kluwer Academic Publishers, Dordrecht

[3] Ueno K, Takasaki K. Toda lattice hierarchy, group representations and systems of differential equations. Adv. Stud. Pure Math., 1984, 4: 1-95

[4] Carlet G, Dubrovin B, Zhang Y. The extended Toda hierarchy. Moscow Mathematical Journal, 2004, 4: 313-332

[5] Carlet G. The extended bigraded Toda hierarchy. J. Phys. A, 2006, 39: 9411-9435

[6] Aoyama S, Kodama Y. Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy. Commun. Math. Phy., 1996, 182: 185-219

[7] Li C Z, He J S, Wu K Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy. J. Math. Phys., 2010, 51: 043514

[8] Li C Z. Solutions of bigraded Toda hierarchy. J. Phys. A, 2011, 44: 255201

[9] Li C Z, He J S, Su Y C. Block type symmetry of bigraded Toda hierarchy. J. Math. Phys., 2012, 53: 013517

[10] Li C Z, He J S. Dispersionless bigraded Toda hierarchy and its additional symmetry. Rev. Math. Phys., 2012, 24: 1230003

[11] Kodama, Y. and Shipman, B., The finite non-periodic Toda lattice: A geometric and topological viewpoint, arXiv: 0805.1389.

[12] Adler M, Moerbeke P V. Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems. Commun. Math. Phys., 1999, 207: 589-620

[13] Adler M, Moerbeke P V. Darboux transforms on band matrices, weights, and associated polynomials. Int. Math. Res. Not., 2001, 18: 935-984

[14] Kodama Y, Ye J. Iso-spectral deformations of general matrix and their reductions on Lie algebras. Commun. Math. Phys., 1996, 178: 765-788

[15] Svinin A K. A class of integrable lattices and KP hierarchy. J. Phys. A, 2001, 34: 10559-10568

[16] Wu Y T, Hu X B. A new integrable differential-difference system and its explicit solutions. J. Phys. A, 1999, 32: 1515-1521