Kadomtsev Petviashvilli Research Articles

Light Meets Water in Nonlocal Media: Surface Tension Analogue in Optics.

Shallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in 2+1 dimensions. We identify an analogue of surface tension in optics, namely, a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a Kadomtsev-Petviashvili (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality. We demonstrate the emergence of robust optical antidark solitons forming Y-, X-, and H-shaped wave patterns, which are approximated by colliding KPII line solitons, similar to those observed in shallow waters.

Asymptotic reductions and solitons of nonlocal nonlinear Schrödinger equations

Asymptotic reductions of a defocusing nonlocal nonlinear Schrödinger model in (3 + 1)-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its far-field, in the form of a variety of Kadomtsev–Petviashvilli (KP) equations for right- and left-going waves, is found. KP models include versions of the KP-I and KP-II equations, in Cartesian and cylindrical geometry. Solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are also predicted to occur. Their nature and stability is determined by a parameter defined by the physical parameters of the original nonlocal system. It is thus found that (dark) anti-dark solitary waves are only supported by a weak (strong) nonlocality, and are unstable (stable) in higher-dimensions. Our analytical predictions are corroborated by direct numerical simulations.

Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation

This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.

On the evolution of perturbations to solutions of the Kadomtsev–Petviashvilli equation using the Benney–Luke equation

The Benney–Luke equation, which arises as a long wave asymptotic approximation of water waves, contains the Kadomtsev–Petviashvilli (KP) equation as a leading-order maximal balanced approximation. The question analyzed is how the Benney–Luke equation modifies the so-called web solutions of the KP equation. It is found that the Benney–Luke equation introduces dispersive radiation which breaks each of the symmetric soliton-like humps well away from the interaction region of the KP web solution into a tail of multi-peaked oscillating profiles behind the main solitary hump. Computation indicates that the wave structure is modified near the center of the interaction region. Both analytical and numerical techniques are employed for working with non-periodic, non-decaying solutions on unbounded domains.

Travelling wave solutions to some important equations of mathematical physics

Exact travelling wave solutions of some nonlinear evolution equations of mathematical physics are obtained by using the mapping method and the extended F -expansion method. It is well known that different types of exact solutions of a given auxiliary ordinary differential equation produce new types of exact solutions to nonlinear evolution equations. Many new exact travelling wave solutions of Zakharov-Kuzentsov (ZK), modified Kadomtsev-Petviashvilli (KP) with square root nonlinearity and modified fifth-order Korteweg-de Vries (KdV) equations are constructed by using mapping method. We also apply the extended F -expansion method to the long-short-wave interaction system and the coupled modified KdV equations. The solutions obtained in this paper include single and combined Jacobi elliptic function solutions, rational solutions and hyperbolic function solutions. In the limiting case, the solitary wave solutions of such equations and systems are also studied.

Interactions of solitary waves under the conditions of the (3 + 1)-dimensional Kadomtsev–Petviashvilli equation

Using an extended mapping method with a linear variable separation process, a new family of the exact solutions of the (3 + 1)-dimensional Kadomtsev–Petviashvilli (KP) equation was derived. By applying the solitary wave solutions, this paper studied some newly localized excitations and the interactions of various solitary waves under the conditions of the (3 + 1)-dimensional KP equation.

A Model Equation for Wavepacket Solitary Waves Arising from Capillary‐Gravity Flows

A model equation governing the primitive dynamics of wave packets near an extremum of the linear dispersion relation at finite wavenumber is derived. In two spatial dimensions, we include the effects of weak variation of the wave in the direction transverse to the direction of propagation. The resulting equation is contrasted with the Kadomtsev–Petviashvilli and Nonlinear Schrödinger (NLS) equations. The model is derived as an approximation to the equations for deep water gravity‐capillary waves, but has wider applications. Both line solitary waves and solitary waves which decay in both the transverse and propagating directions—lump solitary waves—are computed. The stability of these waves is investigated and their dynamics are studied via numerical time evolution of the equation.

Casoratian Solutions to a Non-isospectral Differential-Difference Kadomtsev–Petviashvilli Equation

In this paper, a non-isospectral differential-difference Kadomtsev–Petviashvilli equation (n-DΔKPE) is presented. Then, the Casoratian solutions of the n-DΔKPE are obtained by generalizing Casoratian conditions of the non-isospectral DΔKPE, single-soliton solution is also derived by using Hiorta's method.

The D-bar method, inversion of certain integrals and integrability in 4 + 2 and 3 + 1 dimensions

We first review a method for deriving linear and nonlinear transform pairs, which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d-bar problem. Then, we present two applications of this method: (a) we derive a certain linear transform pair in one dimension, which appears in the characterization of the Dirichlet-to-Neumann map of the Laplace equation in the interior of a convex two-dimensional curvilinear domain. (b) We derive a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy problem of an integrable generalization of the Kadomtsev–Petviashvilli equation in 4 + 2, i.e. in four spatial and two temporal dimensions. The question of reducing this equation form 4 + 2 to 3 + 1 dimensions is also discussed.

Multiple solution profiles to the higher-dimensional Kadomtsev–Petviashvilli equations via Wronskian determinant

Abstract Many types of new exact solutions of (3 + 1)-dimensional KP equation are obtained via an unified Wronskian determinant and three linear partial differential equations, which involve many types of multiple solitary wave solutions, rational solutions, and rational-solitary wave solutions. It is shown that the collisions of the obtained multiple solitary wave solutions are elastic, which implies that (3 + 1)-dimensional KP equation admits multisoliton solutions. Moreover we also give the Wronskian formal solutions of ( n + 1)-dimensional KP equations.

A Generalized Variable-Coefficient Algebraic Method ExactlySolving (3+1)-Dimensional Kadomtsev–PetviashvilliEquation**The project supported by National Natural Science Foundationof China and the Natural Science Foundation of Shandong Province ofChina under Grant No. 2004ZX16

A generalized variable-coefficient algebraic method isapplied to construct several new families of exact solutions of physicalinterest for (3+1)-dimensional Kadomtsev–Petviashvilli (KP) equation. Amongthem, the Jacobi elliptic periodic solutions exactly degenerate to thesoliton solutions at a certain limit condition. Compared with the existingtanh method, the extended tanh method, the Jacobi elliptic function method,and the algebraic method, the proposed method gives new and more generalsolutions.

New doubly periodic and multiple soliton solutions of the generalized (3+1)-dimensional Kadomtsev–Petviashvilli equation with variable coefficients

A generalized variable-coefficient algebraic method is proposed to construct several new families of exact solutions of physical interest for the (3 + 1)-dimensional Kadomtsev–Petviashvilli (KP) equation with variable coefficients. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh method, the extended tanh method, the Jacobi elliptic function method or the algebraic method, the proposed method gives new and more general solutions.

The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW–KP-Type Equations

The Kadomtsev–Petviashvilli (KP) equation,( u t + u x + uu x + u xxx ) x + εu yy =0, (*) arises in various contexts where nonlinear dispersive waves propagate principally along the x-axis, but with weak dispersive effects being felt in the direction parallel to the y-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form ( u t + u x + u p u x + Lu t ) x + εu yy =0, (**) which provides an alternative to Eq.(*) in the same way the regularized long-wave equation is related to the classical Korteweg–de Vries (KdV) equation. The operator L is a pseudo-differential operator in the x-variable, p⩾1 is an integer and ε=±1. After discussing the underlying motivation for the class (**), a local well-posedness theory for the initial-value problem is developed. With assumptions on L and p that include conditions appertaining to models of interesting physical phenomenon, the solutions defined locally in time t are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where L=−∂ x 2 and ε=−1, (*) possesses travelling-wave solutions u( x, y, t)=π c ( x− ct, y) provided c>1 and 0< p<4. It is shown here that these solitary waves are stable for 0<p< 4 3 and c>1 and for 4 3 <p<4 if c>(4 p)/(4+ p). The paper concludes with commentary on extensions of the present theory to more than two space dimensions.

Soliton propagation and interaction on a two-dimensional nonlinear transmission line

Solitons that propagate on a one-dimensional nonlinear dispersive transmission line can be described with the Korteweg–deVries (KdV) equation. In two dimensions, the solitons are described with the Kadomtsev–Petviashvilli (KP) equation. Both equations are derived for the two transmission lines.

Some Special Types of Solitary Wave Solutions for the (3+1)-Dimensional Kadomtsev–Petviashvilli Equation**The project supported by National Natural Science Foundation, "Scaling Plan" of China, and the Natural Science Foundation of Zhejiang Province of China

Using the standard truncated Painlevé analysis, we have obtained some special types of exact explicit solitary wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvilli equations. Usually, one obtains the single solitary wave solution from the Bäcklund transformation related to the truncated Painlevé analysis starting from the trivial vacuum solution. In this letter, we find some special types of the multi-solitary wave solution from the truncated Painlevé analysis and the trivial vacuum solution.

Response to “Comment on ‘Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation’ ” [J. Math. Phys. 40, 3685 (1999)

To verify the correctness of the high order symmetries for the Jimbo–Miwa–Kadomtsev–Petviashvilli system (JMKP) and the potential JMKP (PJMKP) system given in our paper [J. Math. Phys. 36, 3492–3497 (1995)], Ma’s definition of ∂x−1 may be used only for the general nonkernel solutions of the models. For some types of special solutions which are the kernels of some differential operators, one has to use ∂x−1 as the indefinite operator and selected the integral functions appropriately.

Ion acoustic soliton experiments in a plasma

This paper reviews recent laboratory experiments on ion acoustic solitons that are excited, propagate, and interact in a plasma. The solitons can be described with the Korteweg–deVries (KdV) equation, the modified Korteweg–deVries (mKdV) equation, or the Kadomtsev–Petviashvilli (KP) equation. The results should be applicable in non- linear optics experiments.

Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation

The formal series symmetry approach is applied to the conditionally integrable nonlinear evolution equation. A set of infinitely many symmetries for a 3+1-dimensional model, Jimbo–Miwa equation with a conditional equation (Kadomtsev–Petviashvilli equation), are obtained by an explicitly constructable formula. Similar to the usual KP equation, these symmetries constitute a generalized W∞ symmetry algebra.

Multicomponent integrable reductions in the Kadomtsev–Petviashvilli hierarchy

New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy are considered on the basis of Sato’s approach. Within this approach the KP hierarchy is represented by infinite sets of equations for potentials u2,u3,..., of pseudodifferential operators and their eigenfunctions Ψ and adjoint eigenfunctions Ψ*. The KP hierarchy was studied under constraints of the following type (∑ni=1 ΨiΨ*i)x = Sκ,x where Sκ,x are symmetries for the KP equation and Ψi(λi), Ψ*i(λi) are eigenfunctions with eigenvalue λi. It is shown that for the first three cases κ=2,3,4 these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2, Ψ1,...,Ψn, Ψ*1,...,Ψ*n. Bi-Hamiltonian structures for the new hierarchies are presented.

Transverse Diffraction of Nonlinear Waves and Singular Rays

We derive an asymptotic partial differential equation which describes the diffraction of a weakly nonlinear high frequency wave in a direction transverse to its rays. Special cases of this equation are the Zabolotskaya–Khokhlov equation and the Kadomtsev–Petviashvilli equation. We use the equation to analyze singular rays. We also derive generalized Zabolotskaya–Khokhlov equations for sound waves.