D.E. Pelinovsky and Yu.A. Stepanyants
Helical solitons in vector modied Korteweg-de Vries equations
We study existence of helical solitons in the vector modied 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
conrmed by direct numerical simulations.
plasma waves, particle-spring chains, vector modified Korteweg-de Vries equation, helical solitons