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Nonlinear Schrödinger solitons under spatio -temporal forces

Nonlinear Schrödinger solitons under spatio -temporal forces. with Niurka R. Quintero, Sevilla, a nd Alan Bishop, Los Alamos. arXiv : 0907.2438v1. 1. Perturbed NLSE. 2. Collective Coordinate (CC) Theory.

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Nonlinear Schrödinger solitons under spatio -temporal forces

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  1. NonlinearSchrödingersolitonsunderspatio-temporal forces withNiurka R. Quintero, Sevilla, and Alan Bishop, Los Alamos arXiv:0907.2438v1

  2. 1. Perturbed NLSE

  3. 2. Collective Coordinate(CC) Theory Notice: velocityisconstant, althoughforceisperiodic, andphaseisconstant

  4. Ifinitialconditions (IC) nearsteady-statesolutionanddampingbelowcriticalvalue, solitonsalwaysapproachthesteady-statesolution If IC not closetosteady-statesolutionor/anddampingtoo large, thesolitonvanishes, i.e. amplitudeandenergy -> zero, width -> infinity

  5. Case 1b) constant, spatiallyperiodicforce f(x) = exp(iKx), nodamping Stationarysolutions In regionaroundthestationarysolution, thereareoscillatorysolutions Stability? Vakhitov-Kolokolovcriterion valid onlyforstationarysolutions. Try Pego-Weinstein stabilitycriterion : dP/dV > 0, where P and V aresolitonmomentumandvelocity

  6. If „stabilitycurve“ P(V) hasno negative slope, solitonisstable. Notice: unidirectionalmotion, on theaverage, althoughspatialaverageofforcevanishes!

  7. Negative slope: instability

  8. 3. Time dependentforce

  9. 4. Summary Case 1.) constant , spatiallyperiodicforce, nodamping: Oscillatorysolutions, stabilityandlifetimepredictedbyPego-Weinstein-criterion. Unidirectionalmotion , althoughspatialaverageofforcevanishes Withdamping, solitonsapproachsteady-statesolution. Case 2.) acdrivingforce, nodamping: All CCs oscillatewith 3 frequencies: intrinsic, drivingandverylowfrequency Case 3.) biharmonicdrivingforce: Ratcheteffectforunderdampedcase

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