The Korteweg-de Vries Equation (KdV-Equation)
This is the classic example of an equation which exhibits solitons. Methods which are applicable to a large class of equations which exhibit solitons will be derived from the study of this equation and its properties, and we therefore devote a rather large part of this book to this topic.
KeywordsSolitary Wave Wave Packet Poisson Bracket Weak Nonlinearity Single Soliton Solution
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- 2.2R.M. Miura: “The Korteweg -deVries Equation: A Survey of Results”, SIAM Review 18, 412–479 (1976). Presents the exact results known for the KdV, especially the conservation laws. Easy to read, as it is quite detailed, and interesting from the historical point of view. 109 references (to 1975) on the KdV.CrossRefMATHADSMathSciNetGoogle Scholar
- 2.3L.J.F. Broer: “Hidden Hamiltonians of First order Equations, “Physica 79A, 583–596 (1975). A general investigation of how one can describe first-order equations, such as the KdV, by means of a Hamiltonian formalism. In sect. 2.2, we have simply taken the viewpoint of Saletan and Cromer (Theoretical Mechanics Ch. 9, John Wiley and Sons, New York 1971): namely, that a “Hamiltonian mechanics” is defined by the Lie algebra of the Poisson brackets.ADSMathSciNetGoogle Scholar