Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

被引:34
作者
Alber, MS
Luther, GG
Marsden, JE
Robbins, JM
机构
[1] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA
[2] CALTECH, Pasadena, CA 91125 USA
[3] Hewlett Packard Labs, Basic Res Inst Math Sci, Bristol BS12 6QZ, Avon, England
[4] CALTECH, Pasadena, CA 91125 USA
[5] Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England
来源
PHYSICA D | 1998年 / 123卷 / 1-4期
关键词
geometric phase; integrable systems; Lax equation; Lie-Poisson; nonlinear waves; reduction; three wave interaction;
D O I
10.1016/S0167-2789(98)00127-4
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2,1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants, and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context. Copyright (C) 1998 Elsevier Science B.V.
引用
收藏
页码:271 / 290
页数:20
相关论文
共 37 条
[21]   Branches of stable three-tori using Hamiltonian methods in Hopf bifurcation on a rhombic lattice [J].
Kirk, V ;
Marsden, JE ;
Silber, M .
DYNAMICS AND STABILITY OF SYSTEMS, 1996, 11 (04) :267-302
[22]   NORMAL FORMS FOR 3-DIMENSIONAL PARAMETRIC-INSTABILITIES IN IDEAL HYDRODYNAMICS [J].
KNOBLOCH, E ;
MAHALOV, A ;
MARSDEN, JE .
PHYSICA D, 1994, 73 (1-2) :49-81
[23]   INTERACTION OF 3 RESONANT MODES IN A NONLINEAR LATTICE [J].
KUMMER, M .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1975, 52 (01) :64-104
[24]   ON RESONANT CLASSICAL HAMILTONIANS WITH N-FREQUENCIES [J].
KUMMER, M .
JOURNAL OF DIFFERENTIAL EQUATIONS, 1990, 83 (02) :220-243
[25]  
Manakov S.V., 1976, FUNCTIONAL ANAL APPL, V10, P328
[26]  
MARSDEN JE, 1990, MEMOIRS AMS, V436
[27]   RELATIVISTIC SOLITARY-WAVE SOLUTIONS OF THE BEAT-WAVE EQUATIONS [J].
MCKINSTRIE, CJ .
PHYSICS OF FLUIDS, 1988, 31 (02) :288-297
[28]   NONLINEAR DETUNING OF 3-WAVE INTERACTIONS [J].
MCKINSTRIE, CJ ;
CAO, XD .
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 1993, 10 (05) :898-912
[29]   SOLITARY-WAVE SOLUTIONS OF THE GENERALIZED 3-WAVE AND 4-WAVE EQUATIONS [J].
MCKINSTRIE, CJ ;
LUTHER, GG .
PHYSICS LETTERS A, 1988, 127 (01) :14-18
[30]   HOW MUCH DOES THE RIGID BODY ROTATE - A BERRY PHASE FROM THE 18TH-CENTURY [J].
MONTGOMERY, R .
AMERICAN JOURNAL OF PHYSICS, 1991, 59 (05) :394-398