Generalized Kadomtsev Petviashvili Equation Research Articles

Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations

We first review the known mathematical results concerning the KP type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations : blow-up versus long time behavior, stability and instability of solitary waves.

Existence of solitary waves to a generalized kadomtsev-petviashvili equation

In this article, we study the existence of nontrivial solitary waves of a generalized Kadomtsev-Petviashvili equation via variational methods.

A new multi-symplectic scheme for the generalized Kadomtsev–Petviashvili equation

We propose a new scheme for the generalized Kadomtsev–Petviashvili (KP) equation. The multi-symplectic conservation property of the new scheme is proved. Back error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the KPI equation and the KPII equation are presented in detail.

Bäcklund Transformations and Solutions of a Generalized Kadomtsev—Petviashvili Equation

In this paper, the bilinear form of a generalized Kadomtsev—Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Bäcklund transformations are derived.

Stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain

In this work, we are mainly concerned with the existence of stationary solutions for the generalized Kadomtsev-Petviashvili equation in bounded domain in R n 8

Multiple-soliton solutions for a (3 + 1)-dimensional generalized KP equation

The simplified form of the Hirota’s method is used to handle a generalized (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation. Multiple-soliton solutions and multiple singular soliton solutions are formally established. The coefficients of the spatial variables y and z should be of the form ak i and bk i n respectively, where a and b are free parameters and n is finite. The obtained solutions are general and contain other existing solutions.

Large time behavior of solutions for the generalized Kadomtsev-Petviashvili equation

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili (KP) equation { ut +uxxx +σ∂−1 x uyy = −(u)x, (x,y) ∈ R2, t ∈ R, u(0,x,y) = u0(x,y), (x,y) ∈ R2, where σ = 1 or σ = −1 , ∂−1 x = ∫ x −∞ dx′ . Hayashi-Naumkin-Saut [2] have shown asymptotics of solutions for KP equation when ρ 3 and the initial data is sufficiently small and regular. Our aim is to fill the gap of the proof of L∞ time decay of small solutions obtained in [2] and improve their result on the regularity of the data. Mathematics subject classification (2010): 35Q53.

On the generalized Kadomtsev–Petviashvili equation with generalized evolution and variable coefficients

In this Letter, the existence of the solitary wave solution of the Kadomtsev–Petviashvili equation with generalized evolution and time-dependent coefficients will be studied. We use the solitary wave ansätze-method to derive these solutions. A couple of conserved quantities are also computed. Moreover, some figures are plotted to see the effects of the coefficient functions on the propagation and asymptotic characteristics of the solitary waves.

Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation

By using a reductive perturbation method, we derive from Maxwell-Bloch equations a cubic generalized Kadomtsev-Petviashvili equation for ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media without the use of the slowly varying envelope approximation. We calculate the collapse threshold for the propagation of few-cycle spatiotemporal pulses described by the generic cubic generalized Kadomtsev-Petviashvili equation by a direct numerical method and compare it to analytic results based on a rigorous virial theorem. Besides, typical evolution of the spectrum (integrated over the transverse spatial coordinate) is given and a strongly asymmetric spectral broadening of ultrashort spatiotemporal pulses during collapse is evidenced.

Three types of generalized Kadomtsev–Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized (2+1)-dimensional Kadomtsev–Petviashvili (KP) equations are derived from the baroclinic potential vorticity (BPV) equation, including the modified KP (mKP) equation, standard KP equation and cylindrical KP (cKP) equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.

Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev–Petviashvili Equation

In this paper, we investigate the spectral instability of periodic traveling wave solu- tions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.

Asymptotics of the solitary waves for the generalized Kadomtsev-Petviashvili equations

We investigate the asymptotic behaviour of the localized solitarywaves for the generalized Kadomtsev-Petviashvili equations. Inparticular, we compute their first order asymptotics in anydimension $N \geq 2$.

Blow up and instability of solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

Let with p being the ratio of an even to an odd integer. For the generalized Kadomtsev–Petviashvili equation, coupled with Benjamin–Ono equations, in the form it is proved that the solutions blow up in finite time even for those initial data with positive energy. As a by-product, it is proved that for all , the solitary waves are strongly unstable if . This result, even in a special case , improves a previous work by Liu (Liu 2001 Trans. AMS 353 , 191–208) where the instability of solitary waves was proved only in the case of .

On a generalized Kadomtsev–Petviashvili equation with variable coefficients via symbolic computation

Considering the inhomogeneities of media, a generalized Kadomtsev–Petviashvili equation with time-dependent coefficients is hereby investigated with the aid of symbolic computation. The exact analytic one- and two-soliton solutions under certain constraints are obtained by employing the variable-coefficient balancing-act method and Hirota method. Based on its bilinear form, the Lax pair, auto-Bäcklund transformation (in both the bilinear form and the Lax pair form) and nonlinear superposition formula for such an equation are presented. Moreover, some figures are plotted to analyze the effects of the coefficient functions on the stabilities and propagation characteristics of the solitonic waves.

Nontrivial solitary waves to the generalized Kadomtsev–Petviashvili equations

In this note, we study the existence of nontrivial solitary waves for the generalized Kadomtsev–Petviashvili equation in multi-dimensional spaces. Our main tool is a recent linking theorem stated in a paper by Szulkin and Zou [A. Szulkin, W. Zou, Homoclinic orbit for asymptotically linear Hamiltonian systems, J. Funct. Anal. 187 (2001) 25–41].

A new multisymplectic scheme for generalized Kadomtsev-Petviashvili equation

From the multisymplectic Euler box scheme, we derive a new multisymplectic 16-point scheme for the generalized Kadomtsev-Petviashvili equation by eliminating the auxiliary variables. The new scheme is an explicit scheme in sense that it does not need iteration. The results of backward error analysis for the Euler box scheme show that the modified equation associated with the new scheme is not multisymplectic but of a high order approximation to a multisymplectic partial differential equations with a modified structure. Numerical results on one soliton and lump-type solitary waves are reported to illustrate the efficiency of the multisymplectic scheme.

Auto-Bäcklund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients**The project supported by National Natural Science Foundation of China under Grant No. 10072013 and the State Key Basic Research Development Program under Grant No. G1998030600

By using the extended homogeneous balance method, a new auto-Bäcklund transformation(BT) to the generalized Kadomtsev–Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.

Instability of solitary wave solutions to long-wavelength transverse perturbationsin the generalized Kadomtsev-Petviashvili equation with negative dispersion.

The present paper is a more detailed article of the previously published Letter [Phys. Rev. Lett. 84, 3065 (2000)]], which discovered the transversely unstable solitary wave solutions of the generalized Kadomtsev-Petviashvili (GKP) equation with negative dispersion. In addition to detailed explanation of stability analysis to long-wavelength transverse perturbations, numerical calculation of the GKP equation is carried out here to study the transverse stability to perturbations of finite wavelength. The numerical results show that there is a short-wavelength cutoff to the transverse instability. Moreover, we reveal the existence of transversely unstable solitons.

Construction of new soliton-like solutions for the (2 + 1) dimensional KdV equation with variable coefficients

By the application of the extended tanh method and the symbolic computation system-Mathematica, new soliton-like solutions are obtained for the 2D KdV equation with variable coefficients, under a constraint condition. In addition, new soliton-like solutions for the generalized Kadomtsev–Petviashvili equation are obtained in case of degeneracy of the 2D KdV equation.

Nontrivial stationary solutions to GKP equation in bounded domain

In this article, using the Mountain Pass Lemma due to Ambrosetti and Rabinowitz, we obtain the existence of nontrivial stationary solutions of Generalized Kadomtsev–Petviashvili (GKP) equation in a bounded domain with smooth boundary and for superlinear nonlinear term f (u) which satisfies some growth condition. Based on a Pohozaev type variational identity for cylindrical symmetric solution, we obtain the nonexistence of the nontrivial cylindrical symmetric solution for super-critical nonlinearity, i.e. f(u)=|u| p −2 u where .