319 research outputs found
Spectral stability of nonlinear waves in KdV-type evolution equations
This paper concerns spectral stability of nonlinear waves in KdV-type
evolution equations. The relevant eigenvalue problem is defined by the
composition of an unbounded self-adjoint operator with a finite number of
negative eigenvalues and an unbounded non-invertible symplectic operator
. The instability index theorem is proven under a generic
assumption on the self-adjoint operator both in the case of solitary waves and
periodic waves. This result is reviewed in the context of other recent results
on spectral stability of nonlinear waves in KdV-type evolution equations.Comment: 15 pages, no figure
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
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