The sine-Gordon equation with nonlinear delta potential is globally well-posed in H^1_sin x L^2, with stationary waves completely characterized by the parameter q and stability determined by the sign of q.
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Analysis of the sine-Gordon equation with a nonlinear $\delta$-potential
The sine-Gordon equation with nonlinear delta potential is globally well-posed in H^1_sin x L^2, with stationary waves completely characterized by the parameter q and stability determined by the sign of q.