Decomposition of the (2 + 1)- dimensional Gardner equation and its quasi-periodic solutions

To decompose the (2 + 1)-dimensional Gardner equation, an isospectral problem and a corresponding hierarchy of (1 + 1)-dimensional soliton equations are proposed. The (2 + 1)-dimensional Gardner equation is separated into the first two non-trivial (1 + 1)-dimensional soliton systems in the hierarchy, and in turn into two new compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. The Riemann-Jacobi inversion problem is discussed, from which quasi-periodic solutions of the (2 + 1)-dimensional Gardner equation are obtained by resorting to the Riemann theta functions.

[1]  D. Mumford.  Tata Lectures on Theta I , 1982 .

[2]  B. Konopelchenko.  Inverse spectral transform for the (2 + 1)-dimensional Gardner equation , 1991 .

[3]  Xianguo Geng,et al.  Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation , 1999 .

[4]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[5]  B. Konopelchenko,et al.  (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems , 1991 .

[6]  X. Geng,et al.  Classical Integrable Systems Generated Through Nonlinearization of Eigenvalue Problems , 1990 .

[7]  Sergei Petrovich Novikov,et al.  NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES , 1976 .

[8]  Alexey Borisovich Shabat,et al.  The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems , 1987 .

[9]  H. H. Chen,et al.  Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability , 1984 .

[10]  Geng Xian-Guo,et al.  C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy , 1990 .

[11]  E. Previato.  Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation , 1985 .

[12]  C Cao,et al.  NONLINEARIZATION OF THE LAX SYSTEM FOR AKNS HIERARCHY , 1990 .

[13]  Ruguang Zhou,et al.  The finite-band solution of the Jaulent–Miodek equation , 1997 .

[14]  E. Date.  On Quasi-Periodic Solutions of the Field Equation of the Classical Massive Thirring Model , 1978 .

[15]  A. Mikhailov,et al.  Extension of the module of invertible transformations. Classification of integrable systems , 1988 .

[16]  H. Dai,et al.  Algebro-geometric solutions of (2+1)-dimensional coupled modified Kadomtsev–Petviashvili equations , 2000 .

[17]  A. Kirillov.  LOCAL LIE ALGEBRAS , 1976 .

[18]  Fritz Gesztesy,et al.  An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy , 1997, solv-int/9707009.

[19]  Xianguo Geng,et al.  Relation between the Kadometsev–Petviashvili equation and the confocal involutive system , 1999 .

[20]  Decio Levi,et al.  A hierarchy of coupled Korteweg-de Vries equations and the normalisation conditions of the Hilbert-Riemann problem , 1983 .

[21]  Yi Cheng,et al.  The constraint of the Kadomtsev-Petviashvili equation and its special solutions , 1991 .

[22]  M. Gregory Forest,et al.  Spectral theory for the periodic sine‐Gordon equation: A concrete viewpoint , 1982 .

[23]  Boris Konopelchenko,et al.  Some new integrable nonlinear evolution equations in 2 + 1 dimensions , 1984 .

[24]  X. Geng,et al.  From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation , 1999 .

[25]  Xianguo Geng,et al.  Finite-band solutions of the classical Boussinesq–Burgers equations , 1999 .

[26]  F. Gesztesy,et al.  A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy , 1998 .