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Scientific Report 2007-2009
Theoretical physics
T21. Propagation and breaking of weakly nonlinear
and quasi one dimensional waves in Nature
Take any system of nonlinear PDEs i) characterized,
for example, by nonlinearities of hydrodynamic type and
ii) whose linear limit, at least in some approximation,
is described by the wave equation. Then, iii) looking
at the propagation of quasi one dimensional waves and
iv) neglecting dispersion and dissipation, one obtains,
at the second order in the proper multiscale expansion,
the dispersionless Kadomtsev - Petviashvili equation in
n + 1 dimensions (dKP n ): (u t + uu x ) x
+ ∆ ⊥ u = 0, u =
u(x, ⃗y, t), ⃗y = (y 1 , . . . , y n−1 ), ∆ ⊥ = n−1 ∑
∂y 2 i
. Therefore
i=1
dKP n arises in several physical contexts, like acoustics,
plasma physics and hydrodynamics.
We remark that the 1+1 dimensional version of dKP n
is the celebrated Riemann-Hopf equation u t + uu x = 0,
the prototype model in the description of the gradient
catastrophe (or wave breaking) of one dimensional
waves. Therefore a natural question arises: do solutions
of dKP n break and, if so, is it possible to give an analytic
description of such a multidimensional wave breaking?
It was observed long ago that the commutation of multidimensional
vector fields can generate integrable nonlinear
partial differential equations (PDEs) in arbitrary
dimensions. Some of these equations are dispersionless
(or quasi-classical) limits of integrable PDEs, having
dKP 2 as prototype example, they arise in various problems
of Mathematical Physics and are intensively studied
in the recent literature.
We have recently developed the Inverse Spectral
Transform (IST) for 1-parameter families of multidimensional
vector fields, and used it to construct the formal
solution of the Cauchy problem for distinguished examples
of nonlinear PDEs of Mathematical Physics. This
IST and its associated nonlinear Riemann-Hilbert Dressing
scheme turn out to be efficient tools to study also
other relevant properties of the solution space of the
PDE under consideration: i) the characterization of a
distinguished class of spectral data for which the associated
nonlinear RH problem is linearized, corresponding
to a class of implicit solutions of the PDE; ii) the
construction of the longtime behaviour of the solutions
of the Cauchy problem; iii) the possibility to establish
whether or not the lack of dispersive terms in the nonlinear
PDE causes the breaking of localized initial profiles
and, if yes, to investigate in a surprisingly explicit
way the analytic aspects of such a multidimensional wave
breaking.
In this way it was possible to establish that localized
initial data evolving according to dKP 2 generically
break. This exact theory has been recently used to
build a uniform approximation of the solution of the
Cauchy problem for dKP n , for small and localized initial
data, showing that such initial data evolving according
to dKP n break, in the long time regime, if and only if
1 ≤ n ≤ 3; i.e., in physical space. Such a wave breaking
takes place, generically, in a point of the paraboloidal
wave front, and the analytic aspects of it are given explicitly
in terms of the small initial data.
The existence of a critical dimensionality above which
small data do not break has a clear origin, since, in the
model, two terms act in opposite way: the nonlinearity
is responsible for the steepening of the profile, while the
n−1 diffraction channels, represented by the transversal
Laplacian, have an opposite effect; for n = 1, 2, 3 the
nonlinearity prevails and wave breaking takes place (but
at longer and longer time scales, as n increases), while,
for n ≥ 4, the number of transversal diffraction channels
is enough to prevent such phenomenon, in the longtime
regime.
(a)
Figure 1: (a) A detail of the parabolic wave front of dKP 2
at breaking. (b) The compact region in 3D space in which
the solution of dKP 3 is three valued, after breaking
We plan to investigate further such a theory and its
applications, focusing, in particular, on the following
topics. The mathematical aspects of the regularization
of the multidimensional waves evolving according to
dKP n , for n = 2, 3, near breaking, using dispersion
and/or dissipation. The rigorous aspects of the formalism.
The construction of physically interesting explicit
solutions. The applications of this theory to several
physical contexts, like water waves, gas dynamics,
plasma physics and general relativity.
(b)
References
1. S. V. Manakov et al., Theor. Math. Phys. 152, 1004
(2007).
2. S. V. Manakov et al., J. Phys. A41, 055204 (2008).
3. V. Manakov et al., J. Phys. A42, 404013 (2009).
Authors
P. M. Santini
http://solitons.altervista.org/
Sapienza Università di Roma 44 Dipartimento di Fisica