Invariances and Conservation Laws of the Korteweg‐de Vries Equation
Invariances and Conservation Laws of the Korteweg‐de Vries Equation
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DOI：
10.1002/sapm1978592153
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发表时间：
1978-10
影响因子：
2.7
通讯作者：
M. Wadati
中科院分区：
文献类型：
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作者：
M. Wadati
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Taking the Korteweg‐de Vries equation as an example of a soliton system, a connection between conservation laws and infinitesimal transformations (symmetries) is investigated. It is shown that there exist an infinite number of form‐invariant infinitesimal transformations. A one‐to‐one correspondence between conservation laws and infinitesimal transformations is established explicitly. Then it is concluded that the Korteweg‐de Vries equation has an infinite number of conservation laws, each of which corresponds to the symmetry of the system. Two methods of weaving an infinitesimal transformation into a finite one are presented; one of them is the extension of the Lie transformation to the nonlinear case, and the other is a transformation reducible to the Backlund transformation. Furthermore, it is found that Hamiltonian formalism naturally leads to the quantization of the wave field. During the discussions, a number of conjectures and theorems which have been proposed in earlier works are confirmed by using the infinitesimal transformation extensively.