133 research outputs found
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation
The aim of this paper is the accurate numerical study of the KP equation. In
particular we are concerned with the small dispersion limit of this model,
where no comprehensive analytical description exists so far. To this end we
first study a similar highly oscillatory regime for asymptotically small
solutions, which can be described via the Davey-Stewartson system. In a second
step we investigate numerically the small dispersion limit of the KP model in
the case of large amplitudes. Similarities and differences to the much better
studied Korteweg-de Vries situation are discussed as well as the dependence of
the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at
http://www.mis.mpg.de/preprints/index.html
Remarks on the mass constraint for KP type equations
For a rather general class of equations of Kadomtsev-Petviashvili (KP) type,
we prove that the zero-mass (in ) constraint is satisfied at any non zero
time even if it is not satisfied at initial time zero. Our results are based on
a precise analysis of the fundamental solution of the linear part and its anti
-derivative
On critical behaviour in generalized Kadomtsev-Petviashvili equations
An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev\u2013Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlev\ue9 I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves
Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations
Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and
two dimensional Benjamin-Ono (2DBO) equation are considered using parabolic
front initial data. Employing a front tracking type ansatz exactly reduces the
study of DSWs in two space one time (2+1) dimensions to finding DSW solutions
of (1+1) dimensional equations. With this ansatz, the KP and 2DBO equations can
be exactly reduced to cylindrical Korteweg-de Vries (cKdV) and cylindrical
Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which
describe DSW evolution in the cKdV and cBO equations are derived in general and
Riemann type variables are introduced. DSWs obtained from the numerical
solutions of the corresponding Whitham systems and direct numerical simulations
of the cKdV and cBO equations are compared with excellent agreement obtained.
In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO
equations are compared with the cKdV and cBO equations, again with remarkable
agreement. It is concluded that the (2+1) DSW behavior along parabolic fronts
can be effectively described by the DSW solutions of the reduced (1+1)
dimensional equations.Comment: 25 Pages, 16 Figures. The movies showing dispersive shock wave
propagation in Kadomtsev-Petviashvili II and Two Dimensional Benjamin-Ono
equations are available at https://youtu.be/AExAQHRS_vE and
https://youtu.be/aXUNYKFlke
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