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The Cauchy problem for a Schroedinger - Korteweg - de Vries system with rough data  [PDF]
Hartmut Pecher
Mathematics , 2005,
Abstract: The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.
Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds  [PDF]
Alice Mikikits-Leitner,Gerald Teschl
Mathematics , 2009, DOI: 10.1007/978-3-7643-9994-8_7
Abstract: We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg-de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface.
Multi-Lagrangians, Hereditary Operators and Lax Pairs for the Korteweg-de Vries Positive and Negative Hierarchies  [PDF]
Miguel D. Bustamante,Sergio A. Hojman
Physics , 2003, DOI: 10.1063/1.1609035
Abstract: We present an approach to the construction of action principles for differential equations, and apply it to field theory in order to construct systematically, for integrable equations which are based on a Nijenhuis (or hereditary) operator, a ladder of action principles which is complementary to the well-known multi-Hamiltonian formulation. We work out results for the Korteweg-de Vries (KdV) equation, which is a member of the positive hierarchy related to a hereditary operator. Three negative hierarchies of (negative) evolution equations are defined naturally from the hereditary operator as well, in the context of field theory. The Euler-Lagrange equations arising from the action principles are equivalent to the original evolution equation + deformations, which are obtained in terms of the positive and negative evolution vectors. We recognize the Liouville, Sinh-Gordon, Hunter-Zheng and Camassa-Holm equations as negative equations. The ladder for KdV is directly mappable to a ladder for any of these negative equations and other positive equations (e.g., the Harry-Dym and a special case of the Krichever-Novikov equations): a new nonlocal action principle for the deformed system Sinh-Gordon + spatial translation vector is presented. Several nonequivalent, nonlocal time-reparametrization invariant action principles for KdV are constructed. Hamiltonian and Symplectic operators are obtained in factorized form. Alternative Lax pairs for all negative flows are constructed, using the flows and the hereditary operator as only input. From this result we prove that all positive and negative equations in the hierarchies share the same sets of local and nonlocal constants of the motion for KdV, which are explicitly obtained using the local and nonlocal action principles for KdV.
PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation  [PDF]
Bijan Bagchi,Andreas Fring
Physics , 2008, DOI: 10.1088/1751-8113/41/39/392004
Abstract: We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg-de Vries equation. Moreover, we show that these deformations allow for a non-Hermitian Hamiltonian formulation and construct three charges associated to the corresponding flow.
On the stabilization of the Korteweg–de Vries equation  [cached]
Vilmos Komornik
Boletim da Sociedade Paranaense de Matemática , 2010,
Abstract: We consider the Korteweg–de Vries equation on a bounded intervalwith periodic boundary conditions. We prove that a natural mass conserving global feedback exponentially stabilizes the system in all Sobolev norms and we obtain explicit decay rates. The proofs are based on the family of conservation laws for the Korteweg–de Vries equation.
A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation  [PDF]
Iryna Egorova,Gerald Teschl
Mathematics , 2009,
Abstract: Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.
The Korteweg-de Vries Equation  [PDF]
Willy Hereman
Physics , 2013,
Abstract: Two page encyclopedic article about the Korteweg-de Vries equation covering historical perspective, solitary wave and periodic solutions, modern developments, properties and applications, and further reading.
The collaboration between Korteweg and de Vries -- An enquiry into personalities  [PDF]
Bastiaan Willink
Physics , 2007,
Abstract: In the course of the years the names of Korteweg and de Vries have come to be closely associated. The equation which is named after them plays a fundamental role in the theory of non-linear partial differential equations. What are the origins of the doctoral dissertation of De Vries and of the Korteweg-de Vries paper? Bastiaan Willink, a distant relative of both of these mathematicians, has sought to answer these questions. This article is based on a lecture delivered by the author at the symposium dedicated to Korteweg and de Vries at University of Amsterdam in September 2003.
From Agmon-Kannai expansion to Korteweg-de Vries hierarchy  [PDF]
Iosif Polterovich
Physics , 1999,
Abstract: We present a new method for computation of the Korteweg-de Vries hierarchy via heat invariants of the 1-dimensional Schrodinger operator. As a result new explicit formulas for the KdV hierarchy are obtained. Our method is based on an asymptotic expansion of resolvent kernels of elliptic operators due to S.Agmon and Y.Kannai.
The initial-boundary value problem for the Korteweg-de Vries equation  [PDF]
Justin Holmer
Mathematics , 2005,
Abstract: We prove local well-posedness of the initial-boundary value problem for the Korteweg-de Vries equation on the right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators, extending the techniques of Colliander-Kenig (2002).
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