Darboux Transformation and Exact Solutions for Two Dimensional A _2n−1 ⁽²⁾ Toda Equations

Abstract

The Darboux transformation for the two dimensional A _2n−1 ⁽²⁾ Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation.

Keywords

Two dimensional affine Toda equation / Exact solutions / Darboux transformation

References

[1] Alvarez-Fernandez C, Manas M. Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies. Adv. Math., 2013, 240: 132-193

[2] Ao W W, Lin C S, Wei J C. On Toda system with Cartan matrix G(2). Proc. Amer. Math. Soc., 2015, 143: 3525-3536

[3] Aratyn H, Constantinidis C P, Ferreira L A Hirota’s solitons in the affine and the conformal affine Toda models. Nucl. Phys., 1993, B406: 727-770

[4] Bristow R, Bowcock P. Momentum conserving defects in affine Toda field theories. J. High Energy Phys., 2017, 5: 153

[5] Cieśliński J L. Algebraic construction of the Darboux matrix revisited. J. Phys. A, 2009, 42: 404003

[6] Gerdjikov V S. Two-dimensional Toda field equations related to the exceptional algebra g2: Spectral properties of the Lax operators. Theor. Math. Phys., 2012, 172: 1085-1096

[7] Gomes J F, Sotkov G M, Zimerman A H. Nonabelian Toda theories from parafermionic reductions of the WZW model. Ann. Phys., 1999, 274: 289-362

[8] Gu C H, Hu H S, Zhou Z X. Darboux Transformations in Integrable Systems, 2005, Dordrecht: Springer-Verlag

[9] Hu H S. Laplace sequences of surfaces in projective space and two-dimensional Toda equations. Lett. Math. Phys., 2001, 57: 19-32

[10] Leznov A N, Yuzbashjan E A. The general solution of 2-dimensional matrix Toda chain equations with fixed ends. Lett. Math. Phys., 1995, 35: 345-349

[11] Matveev V B. Darboux transformations and the explicit solutions of differential-difference and difference-difference evolution equations. Lett. Math. Phys., 1979, 3: 217-222

[12] Matveev V B, Salle M A. Darboux Transformations and Solitons, 1991, Heidelberg: Springer-Verlag

[13] McIntosh I. Global solutions of the elliptic 2D periodic Toda lattice. Nonlinearity, 1994, 7: 85-108

[14] Mikhailov A V. Integrability of a two-dimensional generalization of the Toda chain. Soviet JETP Lett., 1979, 30: 414-418

[15] Mikhailov A V, Olshanetsky M A, Perelomov A M. Two-dimensional generalized Toda lattice. Commun. Math. Phys., 1981, 79: 473-488

[16] Nie Z H. Solving Toda field theories and related algebraic and differential properties. J. Geom. Phys., 2012, 62: 2424-2442

[17] Nimmo J J C, Willox R. Darboux transformations for the two-dimensional Toda system. Proc. Roy. Soc. London A, 1997, 453: 2497-2525

[18] Nirov K S, Razumov A V. The rational dressing for abelian twisted loop Toda systems. J. High Energy Phys., 2008, 12: 048

[19] Nirov K S, Razumov A V. More non-Abelian loop Toda solitons. J. Phys. A, 2009, 42: 285201

[20] Razumov A V, Saveliev M V. Differential geometry of Toda systems. Commun. Anal. Geom., 1994, 2: 461-511

[21] Rogers C, Schief W K. Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, 2002, Cambridge: Cambridge Univ. Press

[22] Terng, C. L., Geometries and symmetries of soliton equations and integrable elliptic equations, Surveys on Geometry and Integrable Systems (Guest, M., Miyaoka, R. and Ohnita, Y. ed.), Advanced Studies in Pure Mathematics, 51, 2008, 401–488.

[23] Zakharov V B, Mikhailov A V. On the integrability of classical spinor models in two-dimensional space-time. Commun. Math. Phys., 1980, 74: 21-40

[24] Zhou Z X. Darboux transformations and exact solutions of two dimensional A _2n ⁽²⁾ Toda equation. J. Math. Phys., 2005, 46: 033515

[25] Zhou Z X. Darboux transformations of lower degree for two dimensional C _l ⁽¹⁾ and D _l+1 ⁽²⁾ Toda equations. Inverse Problems, 2008, 24: 045016