Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions. (English) Zbl 1218.37090
Summary: In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed by X. Liu et al. [J. Math. Phys. 50, No. 5, 053506 (2009; Zbl 1187.35218)]. General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the \(\tau\)-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 15A04 | Linear transformations, semilinear transformations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Keywords:
extended Toda lattice hierarchy; dressing; reduction; sinh-Gordon; self-consistent sources; solitonCitations:
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