3,152 research outputs found
Embedded Solitons in Lagrangian and Semi-Lagrangian Systems
We develop the technique of the variational approximation for solitons in two
directions. First, one may have a physical model which does not admit the usual
Lagrangian representation, as some terms can be discarded for various reasons.
For instance, the second-harmonic-generation (SHG) model considered here, which
includes the Kerr nonlinearity, lacks the usual Lagrangian representation if
one ignores the Kerr nonlinearity of the second harmonic, as compared to that
of the fundamental. However, we show that, with a natural modification, one may
still apply the variational approximation (VA) to those seemingly flawed
systems as efficiently as it applies to their fully Lagrangian counterparts. We
call such models, that do not admit the usual Lagrangian representation,
\textit{semi-Lagrangian} systems. Second, we show that, upon adding an
infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz,
one can obtain an analytical criterion which (within the framework of VA) gives
a condition for finding \textit{embedded solitons}, i.e., isolated truly
localized solutions existing inside the continuous spectrum of the radiation
modes. The criterion takes a form of orthogonality of the radiation mode in the
infinite tail to the soliton core. To test the criterion, we have applied it to
both the semi-Lagrangian truncated version of the SHG model and to the same
model in its full form. In the former case, the criterion (combined with VA for
the soliton proper) yields an \emph{exact} solution for the embedded soliton.
In the latter case, the criterion selects the embedded soliton with a relative
error .Comment: 10 pages, 1 figur
Physical dynamics of quasi-particles in nonlinear wave equations
By treating the centers of solitons as point particles and studying their
discrete dynamics, we demonstrate a new approach to the quantization of the
soliton solutions of the sine-Gordon equation, one of the first model nonlinear
field equations. In particular, we show that a linear superposition of the
non-interacting shapes of two solitons offers a qualitative (and to a good
approximation quantitative) description of the true two-soliton solution,
provided that the trajectories of the centers of the superimposed solitons are
considered unknown. Via variational calculus, we establish that the dynamics of
the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass
terms. The successful identification of the governing equations of the
(discrete) quasi-particles from the (continuous) field equation shows that the
proposed approach provides a basis for the passage from the continuous to a
discrete description of the field.Comment: 10 pages, 3 figures (6 images); v2: revised and improved the
presentation, updated the references, fixed typos; v3: corrected a few minor
mistakes and typos, version accepted for publication in Phys. Lett.
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