Related papers: Direct delay reductions of the Toda hierarchy
By developing new efficient techniques and using an appropriate fixed point theorem, we derive several new sufficient conditions for the pseudo almost periodic solutions with double measure for some system of differential equations with…
We find a sequence consisting of time dependent evolution vector fields whose time independent part corresponds to the master symmetries for the Toda equations. Each master symmetry decomposes as a sum consisting of a group symmetry and a…
Discrete analogs of the finite and affine Toda field equations are found corresponding to the Lie algebras of series $C_N$ and $\tilde{C_N}$. Their Lax pairs are represented.
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of…
For each one of the Lie algebras $\mathfrak{gl}_{n}$ and $\widetilde {\mathfrak{gl}}_{n}$, we constructed a family of integrable generalizations of the Toda chains characterized by two integers $m_{+}$ and $m_{-}$. The Lax matrices and the…
The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be…
It is shown that some special reduction of infinite 1D Toda lattice gives differential constraints compatible with the Kaup -- Broer system. A family of the travelling wave solutions of the Kaup -- Broer system and its higher version is…
The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as…
The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are…
We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the…
We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and…
We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation,…
It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
Results on continuous dependence on parameters, as well as on regularization, of solutions to linear systems of parabolic partial differential equations of second order with delay are given. One of the main features is that the topology on…
Using Lax-Sato formulation of Manakov-Santini hierarchy, we introduce a class of reductions, such that zero order reduction of this class corresponds to dKP hierarchy, and the first order reduction gives the hierarchy associated with the…
We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with…
The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr\"odinger equations. The well-known…
The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of…