4,461 research outputs found
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
On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks
We study the generalization of the dispersionless Kadomtsev - Petviashvili
(dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model
equation describing the propagation of weakly nonlinear, quasi one dimensional
waves in the absence of dispersion and dissipation, and arising in several
physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear
optics. In 2+1 dimensions and with quadratic nonlinearity, this equation is
integrable through a novel IST, and it has been recently shown to be a
prototype model equation in the description of the two dimensional wave
breaking of localized initial data. In higher dimensions and with higher
nonlinearity, the generalized dKP equations are not integrable, but their
invariance under motions on the paraboloid allows one to construct in this
paper a family of exact solutions describing waves constant on their
paraboloidal wave front and breaking simultaneously in all points of it,
developing after breaking either multivaluedness or single valued discontinuous
shocks. Then such exact solutions are used to build the longtime behavior of
the solutions of the Cauchy problem, for small and localized initial data,
showing that wave breaking of small initial data takes place in the longtime
regime if and only if . At last, the analytic aspects of such a
wave breaking are investigated in detail in terms of the small initial data, in
both cases in which the solution becomes multivalued after breaking or it
develops a discontinuous shock. These results, contained in the 2012 master
thesis of one of the authors (FS), generalize those obtained by one of the
authors (PMS) and S.V.Manakov for the dKP equation in n+1 dimensions with
quadratic nonlinearity, and are obtained following the same strategy.Comment: 31 pages, 11 figure
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
On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation
We establish deep and remarkable connections among partial differential
equations (PDEs) integrable by different methods: the inverse spectral
transform method, the method of characteristics and the Hopf-Cole
transformation. More concretely, 1) we show that the integrability properties
(Lax pair, infinitely-many commuting symmetries, large classes of analytic
solutions) of (2+1)-dimensional PDEs integrable by the Inverse Scattering
Transform method (-integrable) can be generated by the integrability
properties of the (1+1)-dimensional matrix B\"urgers hierarchy, integrable by
the matrix Hopf-Cole transformation (-integrable). 2) We show that the
integrability properties i) of -integrable PDEs in (1+1)-dimensions, ii) of
the multidimensional generalizations of the GL(M,\CC) self-dual Yang Mills
equations, and iii) of the multidimensional Calogero equations can be generated
by the integrability properties of a recently introduced multidimensional
matrix equation solvable by the method of characteristics. To establish the
above links, we consider a block Frobenius matrix reduction of the relevant
matrix fields, leading to integrable chains of matrix equations for the blocks
of such a Frobenius matrix, followed by a systematic elimination procedure of
some of these blocks. The construction of large classes of solutions of the
soliton equations from solutions of the matrix B\"urgers hierarchy turns out to
be intimately related to the construction of solutions in Sato theory. 3) We
finally show that suitable generalizations of the block Frobenius matrix
reduction of the matrix B\"urgers hierarchy generates PDEs exhibiting
integrability properties in common with both - and - integrable
equations.Comment: 30 page
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