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Nonlinear Schrödinger solitons under spatio -temporal forces

This paper explores nonlinear Schrödinger solitons influenced by spatially periodic and time-dependent forces. Utilizing collective coordinate theory, we highlight how solitons approach steady-state solutions under specific initial conditions and damping scenarios. We analyze the stability of these solitons using the Vakhitov-Kolokolov and Pego-Weinstein criteria, focusing on situations with both oscillatory solutions and unidirectional motion. The findings provide insights into the mechanisms governing soliton behavior in the presence of periodic forces, contributing to the broader understanding of soliton dynamics.

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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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