Particle Motions and the Dynamics of Korteweg-de Vries Solutions

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The particlelike behaviour of the soliton solutions of nonlinear evolution equations, raises the question as to whether there exists a representation, in terms of classical particles, which gives quantitative as well as qualitative information about the dynamics of these interacting solitary waveforms. In this paper we establish such a representation for Korteweg-de Vries (KdV) solitons by locating the particles at the poles of the n-soliton solutions when the latter are considered as functions over a complex domain. The representation is shown to be faithful and a detailed analysis is presented for the one- and two-soliton solutions. In these cases the pole motions accurately reflect the behaviour of the solitons ; giving, respectively, a uniform motion and a repulsive interaction. Furthermore, in the case of the two-soliton solutions, the phase shifts calculated from the particle trajectories are the same as those obtained from an asymptotic analysis of the waveforms. Since the KdV is not Galilean invariant, in the strict sense, the mass-spectrum of the pole-particles is obtained from the waveforms using a phenomenologically defined mass density rather than the Hamiltonian density. This mass-spectrum is then used to compute the forces and potentials acting between two solitons and an interpretation of the manner in which this interaction takes place is also given. Finally, a brief discussion of related work on other solutions of the KdV is presented.
理論物理学刊行会の論文
1980-07-25

Particle Motions and the Dynamics of Korteweg-de Vries Solutions

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