540 research outputs found
On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
Non-holonomic deformations of integrable equations of the KdV hierarchy are
studied by using the expansions over the so-called "squared solutions" (squared
eigenfunctions). Such deformations are equivalent to perturbed models with
external (self-consistent) sources. In this regard, the KdV6 equation is viewed
as a special perturbation of KdV equation. Applying expansions over the
symplectic basis of squared eigenfunctions, the integrability properties of the
KdV hierarchy with generic self-consistent sources are analyzed. This allows
one to formulate a set of conditions on the perturbation terms that preserve
the integrability. The perturbation corrections to the scattering data and to
the corresponding action-angle variables are studied. The analysis shows that
although many nontrivial solutions of KdV equations with generic
self-consistent sources can be obtained by the Inverse Scattering Transform
(IST), there are solutions that, in principle, can not be obtained via IST.
Examples are considered showing the complete integrability of KdV6 with
perturbations that preserve the eigenvalues time-independent. In another type
of examples the soliton solutions of the perturbed equations are presented
where the perturbed eigenvalue depends explicitly on time. Such equations,
however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe
The p-Laplace equation in domains with multiple crack section via pencil operators
The p-Laplace equation
\n \cdot (|\n u|^n \n u)=0 \whereA n>0, in a bounded domain \O \subset
\re^2, with inhomogeneous Dirichlet conditions on the smooth boundary \p \O
is considered. In addition, there is a finite collection of curves
\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume
homogeneous Dirichlet boundary conditions} \quad u=0, modeling a multiple
crack formation, focusing at the origin 0 \in \O. This makes the above
quasilinear elliptic problem overdetermined. Possible types of the behaviour of
solution at the tip 0 of such admissible multiple cracks, being a
"singularity" point, are described, on the basis of blow-up scaling techniques
and a "nonlinear eigenvalue problem". Typical types of admissible cracks are
shown to be governed by nodal sets of a countable family of nonlinear
eigenfunctions, which are obtained via branching from harmonic polynomials that
occur for . Using a combination of analytic and numerical methods,
saddle-node bifurcations in are shown to occur for those nonlinear
eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065
Schrödinger operators in the twentieth century
This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics
Must a Hamiltonian be Hermitian?
A consistent physical theory of quantum mechanics can be built on a complex
Hamiltonian that is not Hermitian but instead satisfies the physical condition
of space-time reflection symmetry (PT symmetry). Thus, there are infinitely
many new Hamiltonians that one can construct that might explain experimental
data. One would think that a quantum theory based on a non-Hermitian
Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is
possible to use a previously unnoticed physical symmetry of the Hamiltonian to
construct an inner product whose associated norm is positive definite. This
construction is general and works for any PT-symmetric Hamiltonian. The
dynamics is governed by unitary time evolution. This formulation does not
conflict with the requirements of conventional quantum mechanics. There are
many possible observable and experimental consequences of extending quantum
mechanics into the complex domain, both in particle physics and in solid state
physics.Comment: Revised version to appear in American Journal of Physic
The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
We study two problems. The first one is the similarity problem for the
indefinite Sturm-Liouville operator A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx}
acting in . It is assumed that w,r\in L^1_{\loc}(-b,b) are
even and positive a.e. on .
The second object is the so-called HELP inequality
where the coefficients \tilde{w},\tilde{r}\in L^1_{\loc}[0,b) are
positive a.e. on .
Both problems are well understood when the corresponding Sturm-Liouville
differential expression is regular. The main objective of the present paper is
to give criteria for both the validity of the HELP inequality and the
similarity to a self-adjoint operator in the singular case. Namely, we
establish new criteria formulated in terms of the behavior of the corresponding
Weyl-Titchmarsh -functions at 0 and at . As a biproduct of this
result we show that both problems are closely connected. Namely, the operator
is similar to a self-adjoint one precisely if the HELP inequality with
and is valid.
Next we characterize the behavior of -functions in terms of coefficients
and then these results enable us to reformulate the obtained criteria in terms
of coefficients. Finally, we apply these results for the study of the two-way
diffusion equation, also known as the time-independent Fokker-Plank equation.Comment: 42 page
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