Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons

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Abstract
An exact solution has been obtained for the sine-Gordon equation, \(\frac{\partial^{2}\varphi}{\partial x^{2}}-\frac{\partial^{2}\varphi}{\partial t^{2}}{=}\sin\varphi\), for the case of multiple collisions of solitons with different amplitudes. It is found that solitons behave as if they were particles having apparent attractive forces between them.