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

@article{Badanin2019ThirdorderOW, title={Third-order operators with three-point conditions associated with Boussinesq's equation}, author={Andrey Badanin and Evgeny L. Korotyaev}, journal={Applicable Analysis}, year={2019}, volume={100}, 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.

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## References

SHOWING 1-10 OF 24 REFERENCES

Resonances of third order differential operators

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

Third order operator with periodic coefficients on the real line

- Mathematics
- 2011

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

- Mathematics
- 2005

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

- Mathematics
- 2008

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

- Mathematics
- 2010

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

- Mathematics
- 1918

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

- Mathematics
- 1993

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

- Mathematics
- 2003

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…

ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY

- Mathematics
- 1999

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…