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Unformatted text preview: The Nonlinear Schr¨odinger Equation Eran Bouchbinder December 2003, I. INTRODUCTION The nonlinear Schr¨odinger equation is an example of a universal nonlinear model that describes many physical nonlinear systems. The equation can be applied to hydrodynamics, nonlinear optics, nonlinear acoustics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In this short review I will present a derivation of the nonlinear Schr¨odinger equation in the framework of the general Hamiltonian formalism for nonlinear waves and analyze some of its remarkable features. The modulation instability of plane waves and the appearance of solitary waves will be analyzed. In addition, conservation laws will be discussed. II. DERIVATION OF THE EQUATION Let us consider a spectrally narrow wave packet propagating in an isotropic nonlinear medium. Expanding the Hamiltonian of the system in terms of the amplitude a k of the packet one obtains H = H 2 + H 3 + H 4 , (1) where H 2 = integraltext ω k | a k | 2 d k is the free Hamiltonian, H 3 is the cubic interaction term responsible for three waves processes like confluence and decay and H 4 is the forth order interaction term responsible for four waves processes which include scattering. Our wave packet occupies a narrow region around k in k-space, which means that the amplitude does not vanish only for k similarequal k . Therefore, processes that change the number of waves are non-resonant and all the interaction terms, except the scattering term in H 4 , can be eliminated by a canonical transformation. In that case, Hamilton’s equation i ∂a k ∂t = δ H δa * k , (2) 1 takes the form i ∂a k ∂t- ω k a k = 1 2 integraldisplay T k 123 a * 1 a 2 a 3 δ ( k + k 1- k 2- k 3 ) d k 1 d k 2 d k 3 . (3) Now, since the wave packet is narrow we have k = k + q with | q | greatermuch | k | . Then one can expand the dispersion relation around k to get ω ( k ) = ω + q · v + 1 2 q i q j parenleftBigg ∂ 2 ω ∂k i ∂k j parenrightBigg , (4) where v is the group velocity. In the case of isotropic medium the dispersion relation depends only on the magnitude of k , so we obtain q i q j parenleftBigg ∂ 2 ω ∂k i ∂k j parenrightBigg = q i q j ∂ ∂k j parenleftBigg k i k ∂ω ∂k parenrightBigg = q i q j bracketleftBigg k i k j ω primeprime k 2 + parenleftBigg δ ij- k i k j k 2 parenrightBigg v k bracketrightBigg ≡ q 2 bardbl ω primeprime + q 2 ⊥ v k , (5) where q bardbl and q ⊥ are the projections of q parallel and perpendicular to the direction of prop- agation, respectively. Considering the envelope a k ( t ) = exp (- iω t ) ψ ( q , t ) and substituting into Eq. (3), one obtains bracketleftBigg i ∂ ∂t- q · v- q 2 bardbl ω primeprime 2- q 2 ⊥ v 2 k bracketrightBigg ψ q = T integraldisplay ψ * 1 ψ 2 ψ 3 δ ( q + q 1- q 2- q 3 ) d q 1 d q 2 d q 3 , (6) where T ≡ 1 2 T 0123 . Transforming this equation to the r-space and performing all the δ- functions integrations on the RHS we obtain...
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This note was uploaded on 03/13/2011 for the course ELECTRICAL 605 taught by Professor Mugu during the Spring '08 term at University of Wisconsin - Fond du Lac.
- Spring '08