Kadomtsev-Petviashvili equation

The KP equation is a universal integrable system in two spatial dimensions in the same way that the KdV equation can be regarded as a universal integrable system in one spatial dimension, since many other integrable systems can be obtained as reductions. As such, the KP equation has been extensively studied in the mathematical community in the last forty years. The KP equation is also one of the most universal models in nonlinear wave theory, which arises as a reduction of system with quadratic nonlinearity which admit weakly dispersive waves, in a paraxial wave approximation. The equation naturally emerges as a distinguished limit in the asymptotic description of such systems in which only the leading order terms are retained and an asymptotic balance between weak dispersion, quadratic nonlinearity and diffraction is assumed. The different role played by the two spatial variables accounts for the asymmetric way in which they appear in the equation. Despite their apparent similarity, the two versions of the KP equation differ significantly with respect to their underlying mathematical structure and the behavior of their solutions. Figure 1 shows a two-dimensional localized solution of the KPI equation, known as the lump solution, while Figure 2 shows a contour plot of a resonant two-soliton solution of the KPII equation. Solutions of these KP equations and their properties are discussed in more detail in the following sections.

[1]  Fethi Bin Muhammed Belgacem,et al.  Introducing and Analysing Deeper Sumudu Properties , 2006 .

[2]  Dmitry Pelinovsky,et al.  Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations , 2004, SIAM J. Numer. Anal..

[3]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[4]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[5]  Michio Jimbo,et al.  Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras , 2000 .

[6]  Shu-Ming Sun,et al.  Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem , 2008 .

[7]  Kongqing Yang,et al.  Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature , 2001 .

[8]  Yuji Kodama,et al.  Young diagrams and N-soliton solutions of the KP equation , 2004, nlin/0406033.

[9]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[10]  E. Fan,et al.  Extended tanh-function method and its applications to nonlinear equations , 2000 .

[11]  N. Tzvetkov,et al.  Global low-regularity solutions for Kadomtsev-Petviashvili equation , 2000, Differential and Integral Equations.

[12]  Boris Konopelchenko,et al.  Solitons in Multidimensions: Inverse Spectral Transform Method , 1993 .

[13]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[14]  Malek G. M. Hussain,et al.  TRANSIENT SOLUTIONS OF MAXWELL'S EQUATIONS BASED ON SUMUDU TRANSFORM , 2007 .

[15]  Takahiro Shiota,et al.  Characterization of Jacobian varieties in terms of soliton equations , 1986 .

[16]  Dmitry E. Pelinovsky,et al.  Self-focusing instability of plane solitons and chains of two-dimensional solitons in positive-dispersion media , 1993 .

[17]  R. Hirota.  Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告) , 1976 .

[18]  Vladimir E. Zakharov.  Instability and nonlinear oscillations of solitons , 1975 .

[19]  Javier Villarroel,et al.  On the Discrete Spectrum of the Nonstationary Schrödinger Equation and Multipole Lumps of the Kadomtsev–Petviashvili I Equation , 1999 .

[20]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[21]  Gino Biondini,et al.  On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy , 2003, nlin/0306003.

[22]  M. Boiti,et al.  Inverse scattering theory of the heat equation for a perturbed one-soliton potential , 2002 .

[23]  Alan C. Newell,et al.  Breakdown of Zakharov-Shabat theory and soliton creation , 1977 .

[24]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[25]  Mark J. Ablowitz,et al.  On the evolution of packets of water waves , 1979, Journal of Fluid Mechanics.

[26]  Igor Krichever,et al.  Spectral theory of two-dimensional periodic operators and its applications , 1989 .

[27]  H. Segur,et al.  An Analytical Model of Periodic Waves in Shallow Water , 1985 .

[28]  Ljudmila A. Bordag,et al.  Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction , 1977 .

[29]  V. Prasolov,et al.  Elliptic functions and elliptic integrals , 1997 .

[30]  Sheng Zhang,et al.  APPLICATION OF EXP-FUNCTION METHOD TO HIGH-DIMENSIONAL NONLINEAR EVOLUTION EQUATION , 2008 .

[31]  Ahmet Bekir.  Application of the (G′G)-expansion method for nonlinear evolution equations , 2008 .

[32]  I. Gel'fand,et al.  Fractional powers of operators and Hamiltonian systems , 1976 .

[33]  A. A. Karaballi,et al.  Sumudu transform fundamental properties investigations and applications. , 2006 .

[34]  Javier Villarroel,et al.  The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line , 2002 .

[35]  Wei Wang,et al.  A generalized (G′G)-expansion method for the mKdV equation with variable coefficients , 2008 .

[36]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[37]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water. Part 2. Asymmetric waves , 1995, Journal of Fluid Mechanics.

[38]  J. Satsuma,et al.  Two‐dimensional lumps in nonlinear dispersive systems , 1979 .

[39]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[40]  J. Saut,et al.  Solitary waves of generalized Kadomtsev-Petviashvili equations , 1997 .

[41]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water , 1989, Journal of Fluid Mechanics.

[42]  K. Lonngren,et al.  Nonlinear Waves, Solitons, Shocks, and Chaos , 2010 .

[43]  Dmitry E. Pelinovsky,et al.  Rational solutions of the Kadomtsev–Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution , 1994 .

[44]  Javier Villarroel,et al.  On the Kadomtsev-Petviashvili Equation and Associated Constraints , 1991 .

[45]  Igor Krichever,et al.  Rational solutions of the Kadomtsev — Petviashvili equation and integrable systems of N particles on a line , 1978 .

[46]  John W. Miles,et al.  Diffraction of solitary waves , 1977 .

[47]  Gino Biondini,et al.  Line soliton interactions of the Kadomtsev-Petviashvili equation. , 2007, Physical review letters.

[48]  Gino Biondini,et al.  Soliton solutions of the Kadomtsev-Petviashvili II equation , 2006 .

[49]  Y. Kivshar,et al.  Self-focusing of plane dark solitons in nonlinear defocusing media. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[50]  Yuji Kodama,et al.  Classification of the line-soliton solutions of KPII , 2007, 0710.1456.

[51]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[52]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[53]  J. Nimmo,et al.  Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique , 1983 .

[54]  Fethi Bin Muhammad Belgacem,et al.  Sumudu Applications to Maxwell's Equations , 2009 .

[55]  V. Petviashvili.  Equation of an extraordinary soliton , 1976 .

[56]  M. Boiti,et al.  Scattering transform for nonstationary Schrödinger equation with bidimensionally perturbed N-soliton potential , 2006 .

[57]  Shyam L. Kalla,et al.  ANALYTICAL INVESTIGATIONS OF THE SUMUDU TRANSFORM AND APPLICATIONS TO INTEGRAL PRODUCTION EQUATIONS , 2003 .