For the damped nonlinear Klein-Gordon equation, all 2-solitary waves have opposite signs, the distance between solitons is asymptotically log t, and the initial data form a codimension-2 Lipschitz manifold.
Asymptotic decomposition for nonlinear damped Klein-Gordon equations
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abstract
In this paper, we proved that if the solution to damped focusing Klein-Gordon equations is global forward in time, then it will decouple into a finite number of equilibrium points with different shifts from the origin. The core ingredient of our proof is the existence of the "concentration-compact attractor" which yields a finite number of profiles. Using damping effect, we can prove all the profiles are equilibrium points.
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Description and classification of 2-solitary waves for nonlinear damped Klein-Gordon equations
For the damped nonlinear Klein-Gordon equation, all 2-solitary waves have opposite signs, the distance between solitons is asymptotically log t, and the initial data form a codimension-2 Lipschitz manifold.