Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem

DOI10.1016/0003-4916(89)90178-4zbMath0875.58029OpenAlexW2018596527MaRDI QIDQ1816262

Pavel Winternitz, Luigi Martina

Publication date: 28 January 1997

Published in: Annals of Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0003-4916(89)90178-4

zbMATH Keywords

symmetry groupthree-wave resonant interactionKac-Moody-Virasoro \(\widehat {u} (1)\) algebras

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, (W)-algebras and other current algebras and their representations (81R10) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Invariance and symmetry properties for PDEs on manifolds (58J70)

Related Items

Lie point symmetry and some new soliton-like solutions of the Konopelchenko-Dubrovsky equations, Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations, Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions, Point symmetries of conditionally integrable nonlinear evolution equations, Symmetry reductions of the Skyrme-Faddeev model, Conformally invariant elliptic Liouville equation and its symmetry-preserving discretization, On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system, Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev–Petviashvili hierarchy, Group theoretical analysis of dispersive long wave equations in two space dimensions, Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation, Second harmonic generation: Hamiltonian structures and particular solutions, Method for Solving the Multidimensional n-Wave Resonant Equations and Geometry of Generalized Darboux-Manakov-Zakharov Systems, Symmetries of discrete dynamical systems, On a Schwarzian PDE associated with the KdV hierarchy, Non-linear field dynamics of reacting nuclear, atomic and molecular species, Lie Symmetries, Kac-Moody-Virasoro Algebras and Integrability of Certain (2+1)-Dimensional Nonlinear Evolution Equations, On the symmetry groups of the intrinsic generalized wave and sine–Gordon equations, Symmetries and conditional symmetries of differential-difference equations, Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation

Uses Software

• LIE0

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Automatically determining symmetries of partial differential equations
• Solitons and infinite dimensional Lie algebras
• Infinite dimensional Lie algebras. An introduction
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• Kac-Moody Lie algebras and soliton equations. II: Lax equations associated with \(A_ 1^{(1)}\)
• Nijenhuis G-manifolds and Lenard bicomplexes: A new approach to KP systems
• Monodromy- and spectrum-preserving deformations. I
• Integration of some nonlinear systems of ordinary differential equations
• Equations différentielles du second ordre en Y et du second degré en Y dont l'intégrale générale est à points critiques fixes
• Differential equations with fixed critical points. II
• Differential equations with fixed critical points
• Transformees algébriques des équations du second ordre dont l'intégrale générale est à points critiques fixes
• The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative
• Bi-Hamiltonian formulation of the Kadomtsev–Petviashvili and Benjamin–Ono equations
• Lie symmetries of a generalised non-linear Schrodinger equation. II. Exact solutions
• Lie symmetries of a generalised non-linear Schrodinger equation. III. Reductions to third-order ordinary differential equations
• The Painlevé property for partial differential equations
• The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type
• Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra
• Non-linear superposition formulae for the three-wave resonant interaction via the Hilbert-Riemann problem
• Prolongation structures of nonlinear equations and infinite-dimensional algebras
• KAC-MOODY AND VIRASORO ALGEBRAS IN RELATION TO QUANTUM PHYSICS
• Group analysis of the three-wave resonant system in (2+1) dimensions
• On the infinite-dimensional symmetry group of the Davey–Stewartson equations
• Periodic fixed points of Bäcklund transformations
• Stability and instability of solitary waves of Korteweg-de Vries type
• Solutions of the nonlinear 3-wave equations in three spatial dimensions
• A connection between nonlinear evolution equations and ordinary differential equations of P-type. II
• Similarity Solutions of Some Two-Space-Dimensional Nonlinear Wave Evolution Equations
• The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction
• On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems
• Prolongation structures of nonlinear evolution equations
• Invariants of real low dimension Lie algebras
• Asymptotic Solutions of the Korteweg-deVries Equation
• Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups
• Bäcklund transformation for the resonant three-wave process
• On a unified approach to transformations and elementary solutions of Painlevé equations
• Resonant interactions among capillary-gravity waves