Formation of Quasi-Solitary Wave in Korteweg-de Vries Equation from Initial Wave without Bound State

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Abstract
The Korteweg-de Vries (K-dV) equation is numerically integrated by means of a difference method for the initial rarefactive pulse which has no bound state in the associated eigenvalue equation. It is shown that the compressional pulses produced from the trailing edge of the initial rarefactive pulse have properties similar to solitary waves. The relation among the amplitude, velocity and width agrees well with that of a solitary wave of the K-dV equation, although the amplitude decreases gradually with the time. The result is qualitatively explained by a collisionless shock model.