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D.E. Pelinovsky and Yu.A. Stepanyants

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Helical solitons in vector modied Korteweg-de Vries equations

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**Abstract:**

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.

**Keywords**:

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