413 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
Whitham modulation theory for the Kadomtsev-Petviashvili equation
The genus-1 KP-Whitham system is derived for both variants of the
Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The
basic properties of the KP-Whitham system, including symmetries, exact
reductions, and its possible complete integrability, together with the
appropriate generalization of the one-dimensional Riemann problem for the
Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used
to study the linear stability properties of the genus-1 solutions of the KPI
and KPII equations; it is shown that all genus-1 solutions of KPI are linearly
unstable while all genus-1 solutions of KPII {are linearly stable within the
context of Whitham theory.Comment: Significantly revised versio
Exploring the Dynamics of Nonlocal Nonlinear Waves: Analytical Insights into the Extended Kadomtsev-Petviashvili Model
The study of nonlocal nonlinear systems and their dynamics is a rapidly
increasing field of research. In this study, we take a closer look at the
extended nonlocal Kadomtsev-Petviashvili (enKP) model through a systematic
analysis of explicit solutions. Using a superposed bilinearization approach, we
obtained a bilinear form of the enKP equation and constructed soliton
solutions. Our findings show that the nature of the resulting nonlinear waves,
including the amplitude, width, localization, and velocity, can be controlled
by arbitrary solution parameters. The solutions exhibited both symmetric and
asymmetric characteristics, including localized bell-type bright solitons,
superposed kink-bell-type and antikink-bell-type soliton profiles. The solitons
arising in this nonlocal model only undergo elastic interactions while
maintaining their initial identities and shifting phases. Additionally, we
demonstrated the possibility of generating bound-soliton molecules and
breathers with appropriately chosen soliton parameters. The results of this
study offer valuable insights into the dynamics of localized nonlinear waves in
higher-dimensional nonlocal nonlinear models.Comment: 22 pages, 10 figures; submitted to journa
(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions
We study the problem of gravity surface waves for an ideal fluid model in the
(2+1)-dimensional case. We apply a systematic procedure to derive the
Boussinesq equations for a given relation between the orders of four expansion
parameters, the amplitude parameter , the long-wavelength parameter
, the transverse wavelength parameter , and the bottom variation
parameter . We derived the only possible (2+1)-dimensional extensions
of the Korteweg-de Vries equation, the fifth-order KdV equation, and the
Gardner equation in three special cases of the relationship between these
parameters. All these equations are non-local. When the bottom is flat, the
(2+1)-dimensional KdV equation can be transformed to the Kadomtsev-Petviashvili
equation in a fixed reference frame and next to the classical KP equation in a
moving frame. We have found soliton, cnoidal, and superposition solutions
(essentially one-dimensional) to the (2+1)-dimensional Korteweg-de Vries
equation and the Kadomtsev-Petviashvili equation.Comment: Section 4, with soliton, cnoidal and superposition solutions to
(2+1)-dimensional nonlocal KdV equation, added. In section 5 mistakes
corrected. In Section 6 mistakes correcte
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