Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants

Abstract

For the first two equations of the Volterra lattice hierarchy and the first two equations of its non-autonomous (non-isospectral) extension, we present Riccati systems for functions c j (t), j = 0, 1, . . ., such that an expression in terms of Hankel determinants built from them solves these equations on the right half of the lattice. This actually achieves a complete linearization of these equations of the extended Volterra lattice hierarchy.1 The subscript of u n corresponds to a point on a one-dimensional lattice. We hide away a corresponding subscript of V (1) , for simplicity.

Hungry Lotka–Volterra lattice under nonzero boundaries, block‐Hankel determinant solution, and biorthogonal polynomials

“…Then, based on the molecule solution of the LV lattice under the boundary 𝑎 0 = 0, an extension of the molecule solution of the LV lattice (2) to the case of boundary condition 𝑎 0 ≠ 0 was then explored by Peherstorfer 18 et al Subsequently, Chen et al discussed in Ref. [46] generalizations of some nonisospectral Volterra lattices to nonzero boundary conditions and presented their corresponding molecule solutions. Motivated by these works, a natural question prompted is how to extend the molecule solution of the hLV lattice (1) to a case with nonzero boundary conditions?…”

Bogoyavlensky lattices and generalized Catalan numbers

“…also define an invariant solution submanifold for (35). Moreover, to find u 1 only one equation B 0 = 0 is needed: all other equations are consequences of it and are satisfied automatically, as shown by equation ( 44) which plays the role of a mechanism that provides the induction step.…”

“…We remark that instead of ( 51) one can also use the substitution y j = u 2j+1 + u 2j , z j = u 2j u 2j−1 which also relates the lattice equations ( 52) and (35). For this substitution the calculation is even simpler and we obtain just f 1 = g instead of (60).…”

“…It uses a nonautonomous reduction consistent with the dynamics defined by the lattice equation and related to its master-symmetry. Thus, our problem finds its place among other problems related with the additional symmetry algebra and the string equations, which is a longstudied direction in the theory of integrable equations; we only mention papers [34,35] which are also related to the Volterra lattice. Our reduction turns this lattice equation into a finite-dimensional system and leads to a family of solutions expressed in terms of the Painlevé transcendents, and the additional termination on the half-line leads to special solutions in terms of 1 F 1 .…”

Soliton Solutions for a Nonisospectral Semi-Discrete Ablowitz–Kaup–Newell–Segur Equation

“…Recently, N-soliton solutions to a nonisospectral Ablowitz-Ladik type equation have been investigated by using the Hirota's method and Casorati technique [29]. In addition, based on Hankel type determinants, Chen et al present solutions for a nonisospectral Toda lattice [30] as well as the first and second members in the nonisospectral extended Volterra lattice hierarchy [31].…”