Multi-solitons for nonlinear Klein-Gordon equations (Q2879428)

In this work the authors describe for the first time the existence of multi-soliton solutions for wave type equations by approaching nonlinear Klein-Gordon equations (NLKG). It is well known that a multi-soliton solution is a highly unstable solution, but regardless this instability, the authors are able to construct large mass multi-solitons. To prove this result, they make a whole spectral study of the linearized operators associated to the NLKG equation around the soliton. Due the unstable character of the soliton, they resort to a special functional, composed by a rotation of the standard second-order derivative of the functional for which the soliton is a local minimizer. They prove that the spectrum has three eigenvalues composed by the kernel and two opposite-sign eigenvalues with explicit eigenfunctions. Moreover, they are able to prove the coercivity property for this special functional, concluding that regardless the instability of the soliton, it is possible to construct large mass multi-solitons. Finally, they study the dynamics of small perturbations of multi-solitons by using a nice topological argument.