TY - JOUR
T1 - Strong NLS soliton-defect interactions
AU - Goodman, Roy H.
AU - Holmes, Philip J.
AU - Weinstein, Michael I.
N1 - Funding Information:
RG was supported by NSF DMS-9901897 and Bell Laboratories/Lucent Technologies under the Postdoctoral Fellowship with Industry Program and by NSF DMS-0204881. PH was partially supported by DoE: DE-FG02-95ER25238. We thank D. Pelinovsky for stimulating discussions early in this project. Parts of this paper were presented at ‘Mathematics as a Guide to the Understanding of Applied Nonlinear Problems’, a conference in honor of Klaus Kirchgässner’s 70th birthday, Kloster Irsee, Germany, January 6–10, 2002, and an early version appeared in the informal festschrift for that meeting, edited by H.-J. Kielhöfer, A. Mielke, and J. Scheurle.
PY - 2004/6/1
Y1 - 2004/6/1
N2 - We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.
AB - We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.
KW - Collective coordinates
KW - Hamiltonian systems
KW - Nonlinear scattering
KW - Periodic orbits
KW - Resonant energy transfer
KW - Stable manifolds
KW - Two-mode model
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U2 - 10.1016/j.physd.2004.01.021
DO - 10.1016/j.physd.2004.01.021
M3 - Article
AN - SCOPUS:2342576894
SN - 0167-2789
VL - 192
SP - 215
EP - 248
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 3-4
ER -