KP-II Equation Research Articles

Periodically modulated solitary waves of the CH–KP-I equation

We consider the CH–KP-I equation. For this equation, we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave solutions, i.e. solitary wave solutions to the Camassa–Holm equation, in a dimension-breaking bifurcation. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying the Lyapunov–Iooss theorem.

Lump type solutions: Bäcklund transformation and spectral properties

There are various different ways to obtain traveling waves of lump type(higher order lumps) for the KP-I equation. We propose a general and simple approach to derive them via a Bäcklund transformation. This enables us to establish an explicit connection between those lower energy solutions and higher energy ones. Based on this construction, spectral analysis of the degree 6 solutions is then carried out in details. The analysis of higher energy ones can be done in an inductive way.

Rational Solutions to the KPI Equation as Multi-lumps with a One Degree of Summation

Rational Solutions to the KPI Equation as Multi-lumps with a One Degree of Summation

Conditionally integrable PT -symmetric hierarchies and near-Lax pairs of the KdV and KPII equations

Earlier work is generalized to obtain novel conditionally-integrable hierarchies of various PT -symmetric, nonlinear partial differential equations. The Painlevé Test yields the possible integrable cases. These are labeled by the integer n, the order of the dominant pole in the Laurent expansion for the solution. For the PT -symmetric Korteweg–de Vries (KdV) equation which has been considered earlier, further analysis using the extended homogeneous balance technique gives the near-Lax Pair. Next, a PT -Symmetric hierarchy of (2 + 1) Kadomtsev-Petviashvili equations is considered. The Painlevé Test and Invariant Painlevé analysis in (2 + 1) dimensions yield special solutions and Bäcklund Transformations for the integrable cases.

Two-dimensional vector solitons in Bose-Einstein-condensate mixtures

We derive two decoupled KP-I equations from the system of two-dimensional (2D) Gross-Pitaevskii equations for a two-component Bose-Einstein condensate (BEC), using the multiple-scale expansion method. We produce asymptotic analytical vector-soliton solutions, viz., dark-dark (DD) and dark-antidark (DAD) one-soliton and two-soliton states, by tuning coupling constants and norms of species, and address their evolution numerically under the action of the harmonic-oscillator (HO) trap, in the local-density approximation. We find that shallow single-line DD and DAD solitons are stable, while single-lump DAD solutions (weakly localized truly-2D states) split and lead to nucleation of two half-vortices. We also find that the BEC mixture placed in the HO trap admits stable asymptotic multi-soliton solutions, e.g., two-line DD and DAD solitons and two-lump DD ones, which were not reported before. In particular, the two-lump solutions describe inelastic collisions between the lumps. The analysis developed in this work may also be applied to systems with spin-orbit coupling and gauge fields, which have been realized in atomic BEC.

Low regularity well-posedness of KP-I equations: The three-dimensional case

In this paper low regularity local well-posedness results for the Kadomtsev–Petviashvili–I equation posed in spatial dimension d=3 are proved. Periodic, non-periodic and mixed settings as well as generalized dispersion relations are considered. In the weak dispersion regime these initial value problems show a quasilinear behavior so that bilinear and energy estimates on frequency dependent time scales are used in the analysis.

Linear stability of elastic 2-line solitons for the KP-II equation

The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of 2 2 -line solitons whose line solitons interact elastically each other. Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.

Low regularity well-posedness for KP-I equations: the dispersion-generalized case

We prove new well-posedness results for dispersion-generalized Kadomtsev–Petviashvili I equations in , which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show global well-posedness in . To this end, we combine resonance and transversality considerations with Strichartz estimates and a nonlinear Loomis–Whitney inequality. Moreover, we prove that for small dispersion, the equations cannot be solved via Picard iteration. In this case, we use an additional frequency dependent time localization.

Propagation of regularity for solutions to the KP-I equation

In this paper we prove certain regularity properties of solutions to the IVP for the KP-I equation. Specifically, we prove that if the initial data ϕ has sufficient regularity and if there exists x0∈R such that the initial data also satisfies the additional regularity property that ∂xkϕ∈L2((x0,∞)×R) for certain k, then ∂xku(⋅,t)∈L2((a,∞)×R) for all a∈R, and for all t such that 0<t≤T where T is the existence time of the solution. In particular, regularity on the initial data is transferred to the solution with infinite speed.

Differential Equations for the KPZ and Periodic KPZ Fixed Points

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field for spatially periodic models. For both fields, their multi-point distributions in the space-time domain have been computed recently. We show that for the case of the narrow-wedge initial condition, these multi-point distributions can be expressed in terms of so-called integrable operators. We then consider a class of operators that include the ones arising from the KPZ and the periodic KPZ fixed points, and find that they are related to various matrix integrable differential equations such as coupled matrix mKdV equations, coupled matrix NLS equations with complex time, and matrix KP-II equations. When applied to the KPZ fixed points, our results extend previously known differential equations for one-point distributions and equal-time, multi-position distributions to multi-time, multi-position setup.

Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.

Families of solutions to the KPI equation given by an extended Darboux transformation

Families of solutions to the KPI equation given by an extended Darboux transformation

Stability of KdV solitons with respect to transverse perturbations: Absolute and convective instabilities

We study the stability of one-dimensional solitons propagating in an anisotropic medium. We derived the Kadomtsev-Petviashvili equation for nonlinear waves propagating in an anisotropic medium. By a proper variable substitution this equation reduces either to the KPI or to the KPII equation. In the former case solitons are unstable with respect to the normal modes of transverse perturbations, and in the latter they are stable. We only consider the case when the solitons are unstable. We formulated the linear stability problem. Using the Laplace–Fourier transform, we found the solution describing the evolution of an initial perturbation. Then, using Briggs’ method we studied the absolute and convective instabilities. We found that a soliton is convectively unstable unless it propagates at an angle smaller then critical with respect to a critical direction defined by the condition that the group velocity is parallel to the phase velocity. The critical angle is proportional to the ratio of the dispersion length to the soliton width, which is a small parameter. The coefficient of proportionality is expressed in terms of the phase speed and its second derivative with respect to the angle between the propagation direction and the critical direction. As an example we consider the stability of solitons propagating in Hall plasmas.

The nonlinear superposition between anomalous scattering of lumps and other waves for KPI equation

The nonlinear superposition between anomalous scattering of lumps and other waves for KPI equation

Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

&lt;p style='text-indent:20px;'&gt;A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.&lt;/p&gt;

Lump solutions and interaction solutions for (2 + 1)-dimensional KPI equation

Lump solutions and interaction solutions for (2 + 1)-dimensional KPI equation

New general interaction solutions to the KPI equation via an optional decoupling condition approach

As a kind of analytical exact solutions to the nonlinear evolution equations, the interaction solutions are of great value in the study of the interacting mechanism in nonlinear science. In this paper, an optional decoupling condition approach is proposed for deriving the lump-stripe solutions and lump-soliton solutions to the KPI equation. We derive new and more general solutions to the KPI equation and discuss the link between the two kinds of interaction solutions, which has not been reported before. The interaction solutions to the KPI equation are analyzed and simulated numerically, which show that all the interaction phenomena are completely inelastic. Although we are concerned on the KPI equation in this paper, this approach can be applied to a wide class of nonlinear evolution equations and lays out the framework of deriving the lump-multi-stripe and/or lump-multi-soliton solutions.

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional n -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

Erratum: Abundant solutions of an extended KPII equation combined with a new fourth-order term

Erratum: Abundant solutions of an extended KPII equation combined with a new fourth-order term

Abundant solutions of an extended KPII equation combined with a new fourth-order term

An extension of the KPII equation is studied. Adding a new fourth-order derivative term and some second-order derivative terms, we formulate an extended KPII equation. Different types of solutions of the extended equation are obtained by the Hirota bilinear method, and the presented solutions include soliton solutions, lump solutions and interaction solutions. Their dynamical behaviors are analyzed through plots.