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 $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, we have recently esatablished that, in the nonlocal part of its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$, the formal integral $\partial^{-1}_{x}$ corresponding to the solutions of the Cauchy problem constructed by such an IST is the asymmetric integral $-\int_x^{\infty}dx'$. 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 $f(X,Y)$ over a parabola of the $(X,Y)$ plane in terms of the integrals of $f(X,Y)$ 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 $f(X,Y)$ 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 $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ 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 $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, in this paper we establish the following. 1. The non-local term $\partial_x^{-1}$ arising from its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$ corresponds to the asymmetric integral $-\int_x^{\infty}dx'$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint $\partial_y {\cal M}(y,t)\equiv 0$, where ${\cal M}(y,t)=\int_{-\infty}^{+\infty} \left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2\right]\,dx$. 3. Since no smooth and well-localized initial data can satisfy such constraint at $t=0$, the initial ($t=0+$) 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

The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1-breather Akhmediev solution, whose parameters, different at each RW appearance, are always given in terms of the initial data through elementary functions. If the number of unstable modes is >1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking O(1) corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a uniform in $t$ dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result.Comment: 68 pages, Remark at page 14 and formula (32) of version 1 have been remove

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 $m(n-1)\le 2$. 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 ($S$-integrable) can be generated by the integrability properties of the (1+1)-dimensional matrix B\"urgers hierarchy, integrable by the matrix Hopf-Cole transformation ($C$-integrable). 2) We show that the integrability properties i) of $S$-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 $S$- and $C$- integrable equations.Comment: 30 page