Interaction of Dirac δ-waves in the nonlinear Klein-Gordon equation.

• A review on solving nonlinear hyperbolic systems of PDEs containing Dirac δ measures. • A review on α -product applications to some nonlinear models arising in mathematical physics. • Under certain conditions, δ waves in the Klein-Gordon equation behave like classical solitons in the sine-Gordon equation. The present paper studies the interaction of Dirac δ -waves in models ruled by the nonlinear Klein-Gordon equation u t t − c 2 u x x = ϕ (u) , where c > 0 is a real number and ϕ is an entire function taking real values on the real axis. Such study is made using a product of distributions that both extends the meaning of ϕ (u) for certain distributions u and allows the definition of a solution concept consistent with the classical solution concept. From such study, it emerges that in several nonlinear Klein-Gordon equations Dirac δ -waves behave like classical solitons in the sine-Gordon equation. As particular cases, this work examines the phi-four equation and the sine-Gordon equation. [ABSTRACT FROM AUTHOR]