109,328 research outputs found
Perturbative analysis of generally nonlocal spatial optical solitons
In analogy to a perturbed harmonic oscillator, we calculate the fundamental
and some other higher order soliton solutions of the nonlocal nonlinear
Schroedinger equation (NNLSE) in the second approximation in the generally
nonlocal case. Comparing with numerical simulations we show that soliton
solutions in the 2nd approximation can describe the generally nonlocal soliton
states of the NNLSE more exactly than that in the zeroth approximation. We show
that for the nonlocal case of an exponential-decay type nonlocal response the
Gaussian-function-like soliton solutions can't describe the nonlocal soliton
states exactly even in the strongly nonlocal case. The properties of such
nonlocal solitons are investigated. In the strongly nonlocal limit, the
soliton's power and phase constant are both in inverse proportion to the 4th
power of its beam width for the nonlocal case of a Gaussian function type
nonlocal response, and are both in inverse proportion to the 3th power of its
beam width for the nonlocal case of an exponential-decay type nonlocal
response.Comment: 13 pages, 16 figures, accepted by Phys. Rev.
Symmetries and reductions of integrable nonlocal partial differential equations
In this paper, symmetry analysis is extended to study nonlocal differential
equations, in particular two integrable nonlocal equations, the nonlocal
nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries
equation. Lie point symmetries are obtained based on a general theory and used
to reduce these equations to nonlocal and local ordinary differential equations
separately; namely one symmetry may allow reductions to both nonlocal and local
equations depending on how the invariant variables are chosen. For the nonlocal
modified Korteweg--de Vries equation, analogously to the local situation, all
reduced local equations are integrable. At the end, we also define complex
transformations to connect nonlocal differential equations and
differential-difference equations.Comment: 10 page
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