The Constraint of the Kadomtsev-Petviashvili Equation and Its Special Solutions (original) (raw)
Related papers
Nonlinear quantum-dynamical system based on the Kadomtsev-Petviashvili II equation
The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles. C 2013 AIP Publishing LLC. [http://dx.
Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations
Communications in Nonlinear Science and Numerical Simulation, 2011
We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev-Petviashvili (KP) equations: (u t + 6uu x + u xxx) x ± 3u yy ± 3u zz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Painlevé test. As by-products, the one-soliton and two-soliton solutions and four classes of specific three-soliton solutions are explicitly presented.
Spectral transform and orthogonality relations for the Kadomtsev-Petviashvili I equation
Physics Letters A, 1989
We define a new spectral transform r(k, 1) of the potential u in the time dependent Schrodinger equation (associated to the KPI equation). Orthogonality relations for the sectionally holomorphic eigenfunctions of the Schrodinger equation are used to express the spectral transform f( k, 1) previously introduced by Manakov and Fokas and Ablowitz in terms of r (k, I). The main advantage of the new spectral transform r(k, 1) is that its definition does not require to introduce an additional nonanalytic eigenfunction N. Characterization equations for r(k. 1) are also obtained.
On the Kadomtsev-Petviashvili equation with combined nonlinearities
2021
In this paper, we study the generalized KP equation with combined nonlinearities. First we show the existence of solitary waves of this equation. Then, we consider the associated Cauchy problem and obtain conditions under which solutions are global or blow-up in finite time. We also prove the strong instability of the ground states.
Communications in Nonlinear Science and Numerical Simulation, 2010
Keywords: Jacobi elliptic function Soliton solutions Double periodic solutions Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation a b s t r a c t The periodic wave solutions for Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.
Decomposition of the Modified Kadomtsev–Petviashvili Equation and Its Finite Band Solution
Journal of Nonlinear Mathematical Physics, 2011
The modified Kadomtsev-Petviashvili (mKP) equation is revisited from two 1 + 1-dimensional integrable equations whose compatible solutions yield a special solution of the mKP equation in view of a transformation. By employing the finite-order expansion of Lax matrix, the mKP equation is reduced to three solvable ordinary differential equations (ODEs). The associated flows induced by the mKP equation are linearized under the Abel-Jacobi coordinates on a Riemann surface. Finally, a finite band solution expressed by Riemann-theta functions for the mKP equation is obtained through the Jacobi inversion.
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.
On an elliptic extension of the Kadomtsev–Petviashvili equation
Journal of Physics A: Mathematical and Theoretical, 2014
A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation based on an elliptic Cauchy kernel. This yields a 3 + 1-dimensional lattice system with one of the lattice shifts singled out. The integrability of the lattice system is considered, presenting a Lax representation and soliton solutions. An associated continuous system is also derived, yielding a 3 + 1-dimensional generalisation of the potential KP equation associated with an elliptic curve.
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.,
Explicit construction of solutions of the modified Kadomtsev-Petviashvili equation
Journal of Functional Analysis, 1991
Given a solution of the Kadomtsev-Petviashvili equation we explicitly construct a solution of the modified Kadomtsev-Petviashvili equation related to one another by a generalized Miura transformation. The construction is modeled after a previous treatment of the modified Kortewegde Vries case. As an illustration of our method we derive the soliton solutions of the modified Kadomtsev-Petviashvili equation. 0