6,917 research outputs found
The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
In this paper we apply the formal Inverse Spectral Transform for integrable
dispersionless PDEs arising from the commutation condition of pairs of
one-parameter families of vector fields, recently developed by S. V. Manakov
and one of the authors, to one distinguished class of equations, the so-called
Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs
characterizing a general anti-self-dual conformal structure in neutral
signature, ii) on its first commuting flow, and iii) on some of their basic and
novel reductions. We formally solve their Cauchy problem and we use it to
construct the longtime behavior of solutions, showing, in particular, that
unlike the case of soliton PDEs, different dispersionless PDEs belonging to the
same hierarchy of commuting flows evolve in time in very different ways,
exhibiting either a smooth dynamics or a gradient catastrophe at finite time
An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of
integrable dispersionless multidimensional PDEs is non-local, and the proper
choice of integration constants should be the one dictated by the associated
Inverse Scattering Transform (IST). Using the recently made rigorous IST for
vector fields associated with the so-called Pavlov equation
, we have recently esatablished that, in
the nonlocal part of its evolutionary form , the formal
integral corresponding to the solutions of the Cauchy
problem constructed by such an IST is the asymmetric integral
. In this paper we show that this results could be guessed
in a simple way using a, to the best of our knowledge, novel integral geometry
lemma. Such a lemma establishes that it is possible to express the integral of
a fairly general and smooth function over a parabola of the
plane in terms of the integrals of over all straight lines non
intersecting the parabola. A similar result, in which the parabola is replaced
by the circle, is already known in the literature and finds applications in
tomography. Indeed, in a two-dimensional linear tomographic problem with a
convex opaque obstacle, only the integrals along the straight lines
non-intersecting the obstacle are known, and in the class of potentials
with polynomial decay we do not have unique solvability of the inverse
problem anymore. Therefore, for the problem with an obstacle, it is natural not
to try to reconstruct the complete potential, but only some integral
characteristics like the integral over the boundary of the obstacle. Due to the
above two lemmas, this can be done, at the moment, for opaque bodies having as
boundary a parabola and a circle (an ellipse).Comment: LaTeX, 13 pages, 3 figures. arXiv admin note: substantial text
overlap with arXiv:1507.0820
The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes
The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal
model describing the modulation instability (MI) of quasi monochromatic waves
in weakly nonlinear media, the main physical mechanism for the generation of
rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the
-periodic Cauchy problem for NLS for a generic periodic initial perturbation
of the unstable constant background solution, in the case of unstable
modes. We use matched asymptotic expansion techniques to show that the solution
of this problem describes an exact deterministic alternate recurrence of linear
and nonlinear stages of MI, and that the nonlinear RW stages are described by
the N-breather solution of Akhmediev type, whose parameters, different at each
RW appearence, are always given in terms of the initial data through elementary
functions. This paper is motivated by a preceeding work of the authors in which
a different approach, the finite gap method, was used to investigate periodic
Cauchy problems giving rise to RW recurrence.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1708.00762 and
substantial text overlap with arXiv:1707.0565
Nonlocality and the inverse scattering transform for the Pavlov equation
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of
integrable dispersionless multidimensional PDEs is non-local, and the proper
choice of integration constants should be the one dictated by the associated
Inverse Scattering Transform (IST). Using the recently made rigorous IST for
vector fields associated with the so-called Pavlov equation
, in this paper we establish the
following. 1. The non-local term arising from its
evolutionary form corresponds to the
asymmetric integral . 2. Smooth and well-localized initial
data evolve in time developing, for , the constraint
, where . 3. Since no smooth and well-localized initial
data can satisfy such constraint at , the initial () dynamics of the
Pavlov equation can not be smooth, although, as it was already established,
small norm solutions remain regular for all positive times. We expect that the
techniques developed in this paper to prove the above results, should be
successfully used in the study of the non-locality of other basic examples of
integrable dispersionless PDEs in multidimensions.Comment: 19 page
Geometry of Winter Model
By constructing the Riemann surface controlling the resonance structure of
Winter model, we determine the limitations of perturbation theory. We then
derive explicit non-perturbative results for various observables in the
weak-coupling regime, in which the model has an infinite tower of long-lived
resonant states. The problem of constructing proper initial wavefunctions
coupled to single excitations of the model is also treated within perturbative
and non-perturbative methods.Comment: latex file, 56 pages, 15 figure
A perturbative approach to J mixing in f-electron systems: Application to actinide dioxides
We present a perturbative model for crystal-field calculations, which keeps
into account the possible mixing of states labelled by different quantum number
J. Analytical J-mixing results are obtained for a Hamiltonian of cubic symmetry
and used to interpret published experimental data for actinide dioxides. A
unified picture for all the considered compounds is proposed by taking into
account the scaling properties of the crystal-field potential.Comment: 16 pages + 4 figures; will appear http://prb.aps.or
The Cauchy problem for the Pavlov equation
Commutation of multidimensional vector fields leads to integrable nonlinear
dispersionless PDEs arising in various problems of mathematical physics and
intensively studied in the recent literature. This report is aiming to solve
the scattering and inverse scattering problem for integrable dispersionless
PDEs, recently introduced just at a formal level, concentrating on the
prototypical example of the Pavlov equation, and to justify an existence
theorem for global bounded solutions of the associated Cauchy problem with
small data.Comment: In the new version the analytical technique was essentially revised.
The previous version contained a wrong statement about the solvability of the
inverse problem for large data. This problem remains ope
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