D.E. Pelinovsky and Yu.A. Stepanyants

Helical solitons in vector modi ed Korteweg-de Vries equations


Abstract:
We study existence of helical solitons in the vector modi ed Korteweg-de Vries (mKdV) equations one of which is integrable, whereas another one in non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we obtain helical solitons by the ecient shooting method. The helical solitons arise as a result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is con rmed by direct numerical simulations.

Keywords:
plasma waves, particle-spring chains, vector modified Korteweg-de Vries equation, helical solitons