Third-order operators with three-point conditions associated with Boussinesq's equation

  title={Third-order operators with three-point conditions associated with Boussinesq's equation},
  author={Andrey Badanin and Evgeny L. Korotyaev},
  journal={Applicable Analysis},
  pages={527 - 560}
We consider a non-self-adjoint third-order operator on the interval with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator. 
Spectral asymptotics for the fourth-order operator with periodic coefficients
We consider the self-adjoint fourth-order operator with real 1-periodic coefficients on the unit interval. The spectrum of this operator is discrete. We determine the high energy asymptotics for its
Sharp eigenvalue asymptotics of fourth-order differential operators
Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are
Hill's operators with the potentials analytically dependent on energy


Resonances of third order differential operators
  • E. Korotyaev
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2019
Third order operator with periodic coefficients on the real line
We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness
The inverse periodic spectral theory of the Euler-Bernoulli equation
The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [37], [41], and [38]. A particular case of the inverse problem has been studied in
Zeros of the Green's function for the de la Vallée-Poussin problem
The Green's function for the de la Vallee-Poussin problem where , , , and , is investigated. It is defined in the square , and vanishes at the lines , , , ; it is proved that the orders of its zeros
Even order periodic operators on the real line
We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by
Problems in the theory of ordinary linear differential equations with auxiliary conditions at more than two points
In this paper is considered a differential system consisting of an ordinary linear differential equation and auxiliary conditions involving linearly the values of the solution and its derivatives at
Asymptotic Analysis: Linear Ordinary Differential Equations
This encyclopaedic book describes the developments of the last years in the area of asymptotic methods for linear ODEs and systems in the real and complex domain. Basically all main results and
The periodic Euler-Bernoulli equation
We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem [a(x)u(x)] = λρ(x)u(x), -∞ < x < ∞, where the functions a and p are periodic
We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes