Continued-fraction solutions to the Riccati equation and integrable lattice systems

The integrable Lotka-Volterra (LV) system stands for a prey-predator model in mathematical biology. The discrete LV (dLV) system is derived from a timevariable discretization of the LV system. The solution to the dLV system is known to be represented by using the Hankel determinants. In this paper, we show that, if the entries of the Hankel determinants become m-step Fibonacci sequences at the initial discrete-time, then those are also so at any discrete time. In other words, the m-step Fibonacci sequences always arrange in the entries of the Hankel determinants under the time-variable evolution of the dLV system with suitable initial setting. Here the 2-step and the 3-step Fibonacci sequences are the famous Fibonacci and Tribonacci sequences, respectively. We also prove that one of the dLV variables converges to the ratio of two successive and sufficiently large m-step Fibonacci numbers, for example, the golden ratio in the case where m = 2, as the discrete-time goes to infinity. Some examples are numerically given.

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“…Algebro-geometric solutions of the AL hierarchy were discussed in great detail in [32] (see also [25], [26], [53]). The initial value problem for the half-infinite discrete linear Schrödinger equation and the Schur flow were discussed by Common [17] (see also [18]) using a continued fraction approach. The corresponding nonabelian cases on a finite interval were studied by Gekhtman [24].…”

Section: Introductionmentioning

confidence: 99%

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

Gesztesy¹,

Holden²,

Michor³

2010

Discrete &Amp; Continuous Dynamical Systems - A

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We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1 + 1)-dimensional completely integrable soliton equations of differential-difference type.

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Continued-fraction solutions to the Riccati equation and integrable lattice systems

Cited by 20 publications

References 9 publications

Boundary value problems for integrable equations compatible with the symmetry algebra

Boundary value problems for integrable equations compatible with the symmetry algebra

On m-step Fibonacci sequence in discrete Lotka-Volterra system

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

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