Method Of Inverse Spectral Problem Research Articles

Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.

Zadacha Koshi dlya nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsiy

The inverse spectral problem method is applied to finding a solution of the Cauchyproblem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded termsand a derivation of an analog of Dubrovin’s system of differential equations are proposed. Itis shown that the sum of a uniformly convergent function series constructed by solving theDubrovin system of equations and the first trace formula actually satisfies the loaded nonlinearKorteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodicanalytic function, then the solution of the Cauchy problem is a real analytic function in thevariable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initialfunction, then the number π/n is the period for solving the Cauchy problem with respect to thevariable x.

Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions

In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a π \pi -periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable x x ; and if the number π / 2 \pi /2 is a period (antiperiod) of the initial function, then the number π / 2 \pi /2 is a period (antiperiod) in the variable x x of the solution of the Cauchy problems for the Hirota equation.

Periodic solutions of the modified Korteweg–de Vries equation in hemodynamics

In this paper, we consider the periodic solution of the modified Korteweg-de Vries equation used for studies of hemodynamic processes. It is shown that the modified Korteweg-de Vries equation can be integrated by the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the modified Korteweg-de Vries equation is determined. The obtained results substantiate the applicability of the inverse problem method for solving the modified Korteweg-de Vries equation for studying the laws of hemodynamics.

Prediction of Multidimensional Time Series by Method of Inverse Spectral Problem

Prediction of Multidimensional Time Series by Method of Inverse Spectral Problem

Direct and inverse problems for vector logarithmic potentials with external fields

We consider extremal problems for the energy of the logarithmic potential with external fields closely related with the inverse spectral problem method. The method is based on the relations between the external field and the supports of the equilibrium measures which were discovered in the pioneering papers of Rakhmanov, Saff, Mhaskar and Buyarov (RSMB-method). We propose a generalization of the RSMB-method for the vector of measures with matrix of interaction between components.

Inverse spectral problem for finite Jacobi matrices with zero diagonal

In this study, the necessary and sufficient conditions for solvability of an inverse spectral problem about eigenvalues and normalizing numbers for finite-order real Jacobi matrices with zero diagonal elements are established. An explicit procedure of reconstruction of the matrix from the spectral data consisting of the eigenvalues and normalizing numbers is given. Numerical examples and error analysis are provided to demonstrate the solution technique of the inverse problem. The results obtained are used to justify the solving procedure of the finite Langmuir lattice by the method of inverse spectral problem.

Solution of the finite complex Toda lattice by the method of inverse spectral problem

We show that the finite Toda lattice with complex-valued initial data can be integrated by the method of inverse spectral problem. For this goal spectral data for complex Jacobi matrices are introduced and an inverse spectral problem with respect to the spectral data is solved. The time evolution of the spectral data for the Jacobi matrix associated with the solution of the Toda lattice is computed. Using the solution of the inverse spectral problem with respect to the time-dependent spectral data we reconstruct the time-dependent Jacobi matrix and hence the desired solution of the finite complex Toda lattice.

The Nonlinear Schrödinger Equation with a Self-consistent Source in the Class of Periodic Functions

In this work the method of inverse spectral problem is applied to the integration of the nonlinear Schrodinger equation with a self-consistent source in the class of periodic functions.

The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions

We use the inverse spectral problem method to integrate the Korteweg-de Vries equation with a self-consistent source in the class of periodic functions.

An inverse spectral problem for complex Jacobi matrices

We introduce the concept of generalized spectral function for finite order complex Jacobi matrices and solve the inverse problem with respect to the generalized spectral function. The results obtained can be used for solving of initial-boundary value problems for finite nonlinear Toda lattices with the complex-valued initial conditions by means of the inverse spectral problem method.

Completely integrable nonlinear dynamical systems of the Langmuir chains type

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Pade approximations, and on the Sturm-Liouville method for finite difference equations is used.

Integration of non-abelian langmuir type lattices by the inverse spectral problem method

Integration of non-abelian langmuir type lattices by the inverse spectral problem method

Nonisospectral Flows on Semi-infinite Jacobi Matrices

It is proved that if the spectrum and spectral measure of a semi-infinite Jacobi matrix L(t) change appropriately, then L(t) satisfies a generalized Lax equation of the form ˙ L(t) = ( L(t),t)+[L(t),A(L(t),t)], where ( ,t ) is a polynomial with t-dependent coecients and A(L(t),t) is a skew-symmetric matrix which is determined by the evolution of the spectral data. Such an equation is equivalent to a wide class of generalized Toda lattices. The theory of Jacobi matrices gives rise to the procedure of solution of the corresponding Cauchy problem by the inverse spectral problem method. The linearization of this nonlinear equation in terms of the moments is established.

NonAbelian nonlinear lattice equations on finite interval

We apply the inverse spectral problem method to the class of non-Abelian nonlinear lattice equations on the finite interval. The integrable discrete nonlinear Schrodinger and discrete modified Korteveg-de Vries equations are considered as examples. In the latter case the large time asymptotics for solutions are found.