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This article is cited in 15 scientific papers (total in 15 papers)
Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices
Yu. M. Berezanskiia, A. A. Mokhon'kob a Institute of Mathematics, Ukrainian National Academy of Sciences
b National Taras Shevchenko University of Kyiv
Abstract:
The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution $u(t)$, $t\in[0,\infty)$, of the problem, one assigns a self-adjoint
semi-infinite Jacobi matrix $J(t)$ whose spectral measure $d\rho(\lambda;t)$ undergoes simple evolution in time $t$. The solution of the Cauchy problem goes as follows. One writes out the spectral measure $d\rho(\lambda;0)$ for the initial value $u(0)$ of the solution and the corresponding Jacobi matrix $J(0)$ and then computes the time evolution $d\rho(\lambda;t)$ of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi
matrix $J(t)$ from $d\rho(\lambda;t)$ 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 $d\rho(\zeta;t)$ 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.
Keywords:
block Jacobi matrix, generalized eigenvector, spectral representation, Toda lattice.
Received: 29.05.2007
Citation:
Yu. M. Berezanskii, A. A. Mokhon'ko, “Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices”, Funktsional. Anal. i Prilozhen., 42:1 (2008), 1–21; Funct. Anal. Appl., 42:1 (2008), 1–18
Linking options:
https://www.mathnet.ru/eng/faa2886https://doi.org/10.4213/faa2886 https://www.mathnet.ru/eng/faa/v42/i1/p1
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Abstract page: | 850 | Full-text PDF : | 301 | References: | 72 | First page: | 16 |
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