Bilinear Backlund, Lax Pairs, Lump waves and Soliton interaction of (2+1)-dimensional non-autonomous Kadomtsev-Petviashvili equation (original) (raw)
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Line Soliton Interactions of the Kadomtsev-Petviashvili Equation
Physical Review Letters, 2007
We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4ut+6uux+3uxxx)x+uyy=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y→∞ coincide with those of the N asymptotic line solitons as y→-∞. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.
Radiophysics and Quantum Electronics, 1987
The structure of steady-state two-dimensional solutions of the soliton type with quadratic and cubic nonlinearities and power-law dispersion is analyzed numerically. It is shown that steadily coupled two-dimensional multisolitons can exist for positive dispersion in a broad class of equations, which generalize the Kadomtsev-Petviashvili equation. i. Kadomtsev and Petviashvili [i] have derived an equation that generalizes the wellknown Korteweg-de Vries equation and describes quasiplanar disturbances in a quadratically nonlinear medium with weak dispersion. The basic approximation used in [i] is the assumption that the scale of the wave field in the direction of motion is much smaller than the scale in the transverse direction. Clearly, the same approximation can also be used to describe disturbances in other media having different types of nonlinearity and dispersion (see, e.g.,
Darboux transformation for soliton solutions of the modified Kadomtsev-Petviashvili-II equation
2018
Soliton solutions as far as hyperbolic cosines to the modified Kadomtsev–Petviashvili II equation are displayed. The behaviour of each line soliton in the far region can be characterized analytically. It is revealed that under certain conditions, there may appear an isolated bump in the interaction centre, which is much higher in peak amplitude than the surrounding line solitons, and the more line solitons interact, the higher isolated bump will form. These results may provide a clue to generation of extreme high-amplitude waves, in a reservoir of small waves, based on nonlinear interactions between the involved waves.
Physica Scripta, 2012
We give an introduction to a new direct computational method for constructing multiple soliton solutions to nonlinear equations with variable coefficients in the Kadomtsev-Petviashvili (KP) hierarchy. We discuss in detail how this works for a generalized (3 + 1)-dimensional KP equation with variable coefficients. Explicit soliton, multiple soliton and singular multiple soliton solutions of the equation are obtained under certain constraints on the coefficient functions. Furthermore, the characteristic-line method is applied to discuss the solitonic propagation and collision under the effect of variable coefficients.
Filomat
The Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) model equations as a water wave model, are governing equations, for fluid flows, describes bidirectional propagating water wave surface. The soliton solutions for (2+1) and (3+1)-Dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equations have been extracted. The solitary wave ansatz method are adopted to approximate the solutions. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of the problem.
A Soliton Solution for the Kadomtsev–Petviashvili Model Using Two Novel Schemes
Symmetry
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili (KP) model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation (DT). The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the DT is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed an...
EDITORIAL: “Solitons, Integrability, Nonlinear Waves: Theory and Applications”
The European Physical Journal Plus, 2021
Nonlinear waves have long been at the research focus of both physicists and mathematicians, in diverse settings ranging from electromagnetic waves in nonlinear optics to matter waves in Bose-Einstein condensates, from Langmuir waves in plasma to internal and rogue waves in hydrodynamics. The study of physical phenomena by means of mathematical models often leads to nonlinear evolution equations known as integrable systems. One of the distinguished features of integrable systems is that they admit soliton solutions, i.e., stable, localized traveling waves which preserve their shape and velocity in the interaction. Other fundamental properties of integrable systems are their universal nature, and the fact that they can be effectively linearized, e.g., via the Inverse Scattering Transform (IST), or reduced to appropriate Riemann-Hilbert problems. Moreover, explicit solutions can often be derived by the Zakharov-Shabat dressing method, by Bäcklund or Darboux transformations, or by Hirota's bilinear method. Prototypical examples of such integrable equations in 1+1 dimensions are the nonlinear Schrödinger (NLS) equation and its multicomponent generalizations, the sine-Gordon equation, the Korteweg-de Vries (KdV) and the modified KdV equations, the Kulish-Sklyanin model, etc. The most notable examples of integrable systems in 2+1-dimensions are the Kadomtsev-Petviashvili (KP) equations, and the Davey-Stewartson equations. The aim of this special issue is to present the latest developments in the theory of nonlinear waves and integrable systems, and some of their applications. Below, we briefly outline the contributions to the present Focus Point (FP) summarizing their achievements in nonlinear wave phenomena and integrable systems. Some open problems and questions are also identified.
Romanian Reports in Physics, 2013
This paper studies the (3+1)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity that apperas in the study of multi-component plasmas. The solutions are obtained by several methods such as modified F-expansion method, exp-function method, / G G ′ expansion method, ansatz method, traveling wave hypothesis, the improved Jacobi's elliptic function method and Lie symmetry analysis. These method lead to several closed form exact solutions. Some of these solutions are topological, non-topological and singular solitons, cnoidal, snoidal waves. It is also shown that in the limiting case, these doubly periodic functions lead to singular periodic functions, complexitons and linear waves. The domain restrictions are also identfified in order for the soliton solutions to exist.
Bharathidasan University, India: Ph.D. Thesis, 2015
Solitons are very important nonlinear entities in the recent years which find multifaceted applications in different branches of science, engineering, and technology, due to their ability to propagate over extraordinary distances without any loss of energy and remarkable stability under collisions. These solitons give rise to several interesting features in the associated nonlinear dynamical systems. Especially, the multicomponent solitons show distinct propagation characteristics and posses fascinating energy sharing collisions which are not possible in scalar (single component) solitons. In this thesis, we consider a set of multicomponent nonlinear dynamical systems, such as multicomponent Yajima-Oikawa equations in (1+1)-dimension, long-wave– short-wave resonance interaction equations in (2+1)-dimensions, coherently coupled nonlinear Schro ̈dinger equations in the presence of four-wave mixing nonlinearities (with same type as well as opposite signs for nonlinearities), and three-coupled Gross-Pitaevskii equations with spin-mixing nonlinearities, arising in the context of nonlinear optics and Bose-Einstein condensates. After studying the integrability nature of these equations, we construct explicit multicomponent soliton solutions which supports a variety of profiles like single-hump, double-hump, flat-top, dark (hole), and gray solitons for different choices of parameters. Our studies on bright soliton collisions reveal different types of energy-sharing and energy-switching collisions in addition to their elastic collisions. Studies on special solutions like bound states and resonant solitons show periodic oscillations which can be controlled by altering the polarization parameters. But, the dark solitons admit only elastic collisions. Our results can find interesting applications in different context of nonlinear science.