In [1] the authors have considered N = 2 supersymmetric Landau-Ginzburg theories and have showed that in many cases the metric for supersymmetric ground states for special deformations of this metric satisfies the certain system of PDE’s, such as Toda equations. For further development related to the theory see [2], [3]. In the note [4] two new examples of such theories were considered. The pur...

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Pois...

with potential V (x) = γe + V1x + V2 and γ, δ, V1, V2 ∈ R constants. The Toda lattice has been introduced by Toda [12] and studied extensively in the sequel. It is an FPU lattice, i.e. a Hamiltonian system of particles in one space dimension with nearest neighbor interaction. Models of this type have been studied by Fermi-Pasta-Ulam [FPU]. In numerical experiments they found recurrent features ...

In this paper we establish for the first time the bi-Hamiltonian nature of the Toda lattices corresponding to classical simple Lie groups. These are systems that generalize the usual finite, non-periodic Toda lattice (which corresponds to a root system of type An). This generalization is due to Bogoyavlensky [2]. These systems were studied extensively in [10] where the solution of the system wa...

system (1.1) is completely integrable, which is well-known in integrable systems theory. The Liouville equation and the Toda system arise in many physical models. In Chern-Simons theories, the Liouville equation is closely related to Abelian models, while the Toda system is related to non-Abelian models. See for instance the books [10] and [24] and the references therein. Though the Liouville e...

We consider the following Toda system

Toda in his celebrated paper showed that the polynomial-time hierarchy is contained in P#P. We give a short and simple proof of the first half of Toda’s Theorem that the polynomial-time hierarchy is contained in BPP⊕P. Our proof uses easy consequences of relativizable proofs of results that predate Toda. For completeness we also include a proof of the second half of Toda’s Theorem. ACM Classifi...

In a certain class of differential-difference equations for dissipative systems, we show that hyperbolic tangent model is the only the nonlinear system of equations which can admit some particular solutions of the Toda lattice. We give one parameter family of exact solutions, which include as special cases the Toda lattice solutions as well as the Whitham’s solutions in the Newell’s model. Our ...

We study the general form of the equations for isotropic single-scalar, multi-scalar and dyonic p-branes in superstring theory and M-theory, and show that they can be cast into the form of Liouville, Toda (or Toda-like) equations. The general solutions describe non-extremal isotropic p-branes, reducing to the previously-known extremal solutions in limiting cases. In the non-extremal case, the d...

We find new solutions, including soliton-like ones, for a special case of nonAbelian loop Toda equations associated with complex general linear groups. We use the method of rational dressing based on an appropriate block-matrix representation suggested by the Z-gradation under consideration. We present solutions in a form of a direct matrix generalization of the Hirota’s soliton solution alread...