Integration of semi-infinite Toda chain in some class of unbounded solutions

Szczepanik, Mariusz

doi:10.1016/s0034-4877(97)88005-6

Cited by 5 publications

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References 9 publications

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“…(In the case of a Toda lattice, this spectrum does not depend on t.) In [35], consideration was given to the situation in which the Jacobi matrix J(t) generates an unbounded (but self-adjoint) operator in 2 .…”

Section: Dedicated To the Fond Memory Of M G Krein On The Occasion mentioning

confidence: 99%

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Berezansky

Mokhon’ko

2008

Funct Anal Its Appl

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The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈ [0, ∞), of the problem, one assigns a self-adjoint semiinfinite Jacobi matrix J(t) whose spectral measure dρ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dρ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi matrix J(0) and then computes the time evolution dρ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix J(t) from dρ(λ; t) and hence finds the desired solution u(t).In the present paper, this approach is generalized to the case in which the role of J(t) is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dρ(ζ; t) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.To integrate the Cauchy problem with respect to time t for the nonlinear Korteweg-de Vries equation, there exists a well-known inverse spectral problem method for the self-adjoint SturmLiouville equation on the half-line. This method was put forward and developed in the classical papers by I. M. Gelfand, B. M. Levitan, V. A. Marchenko, and M. G. Krein (for its presentation, see [1]). Using some results for a finite Toda lattice (see [2], [3]), one of the authors [4]-[6]suggested a similar integration method for the Cauchy problem with respect to t in the case of a nonlinear difference equation, a half-infinite Toda lattice, on the basis of the simpler spectral theory for the difference analog of the Sturm-Liouville equation, a semi-infinite self-adjoint Jacobi matrix. We point out that in both cases use was made of self-adjoint operators.The main objective of the present paper is to show that a similar integration theory for the corresponding classes of nonlinear difference equations can be developed on the basis of the recently constructed spectral theory of normal semi-infinite Jacobi matrices (see [7], [8]). In this approach, there necessarily arise matrix (non-Abelian) nonlinear equations; this is related to the stringent property that normal tridiagonal matrices are necessarily matrices of block Jacobi structure with (as a rule) increasing finite dimensions of the blocks rather than to the desire for generalizations.We now give some remarks on the spectral theory of block self-adjoint Jacobi matrices. This theory was created under the influence of the note [9] by M. G. Krein, published in 1949. In particular, it was developed in [10]-[12] and found application in the integration of the Cauchy problem for some classes of matrix difference equations [13]- [16]. (Spectral results of some type that are related to the general block Jacobi matrices and developed in [17]...

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Section: Dedicated To the Fond Memory Of M G Krein On The Occasion mentioning

confidence: 99%

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Berezansky

Mokhon’ko

2008

Funct Anal Its Appl

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“…Подсчитаем вклад вȧ n третьего слагаемого в (35). Для этого воспользуемся первым и вторым из равенств (9) (при z = ζ иz =ζ ), которым удовлетворяет вектор P n;· (ζ; t).…”

Section: теорема 3 пусть рассматриваемая блочная нормальная матрица unclassified

“…Найдем вклад первого слагаемого в (35). Для этого продифференцируем по t второе из равенств (9) (приz =ζ ).…”

Section: теорема 3 пусть рассматриваемая блочная нормальная матрица unclassified

“…Аналогично подсчитывается вклад второго слагаемого в (35). При этом нуж-но продифференцировать по t первое из равенств (9) при z = ζ с заменой n на n + 1.…”

Section: теорема 3 пусть рассматриваемая блочная нормальная матрица unclassified

“…В [35] был рассмотрен случай, ко-гда якобиева матрица J(t) порождает неограниченный (но самосопряженный) оператор в 2 .…”

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Integration of semi-infinite Toda chain in some class of unbounded solutions

Cited by 5 publications

References 9 publications

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Интегрирование Некоторых Дифференциально-Разностных Нелинейных Уравнений С Помощью Спектральной Теории Блочных Якобиевых Нормальных Матриц

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