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## translations in space: x

translations in space: x → x + ∆x translations in time: t → t + ∆t As in the forced oscillator case, the translations imply that the evolution equation for A is equivariant under A → Ae iφ for arbitrary φ ⇒ expect an evolution equation of the form Action of the reflections: ∂TA = aA + c|A| 2 A + v∂XA + d∂ 2 XA + . . . (33) • under spatial reflections (and under reflections in time) a right-traveling wave is transformed into a left-traveling wave • (32) includes only a right-traveling wave: pure spatial reflections cannot be represented within the class of functions (32) • combined reflections in time and space, however, map a right-traveling wave again into a right-traveling wave ⇒ they have a simple action on the amplitude A in the ansatz (32) x → −x combined with t → −t induces T2 → −T2, X → −X, A → A ∗ Applied to the general evolution equation (33) the transformation yields taking the complex conjugate implies and Thus: −∂TA ∗ = aA ∗ + c|A| 2 A ∗ − v∂XA ∗ + d∂ 2 X A∗ + . . . −∂TA = a ∗ A + c ∗ |A| 2 A − v ∗ ∂XA + d ∗ ∂ 2 XA + . . . a = −a ∗ d = −d ∗ ∂TA = v∂XA + i 1 d 2 2ω Going into a moving frame X → X + vT one gets Notes: c = −c ∗ dq2 ∂2 XA + i 1 4 v = v ∗ ω 2 0 ω |A|2 A ∂TA = i 1 d 2 2ω dq2 ∂2 1 ω XA + i 4 2 0 ω |A|2A • this equation is the nonlinear Schrödinger equation (NLS) • the NLS is the generic description for small-amplitude waves in non-dissipative media • in a multiple-scale analysis one has to introduce actually two slow times T1 = ǫt and T2 = ǫ 2 t and gets two non-trivial solvability conditions: at O(ǫ 2 ) one gets ∂T1A = v∂XA and O(ǫ 3 ) one gets the NLS. 76

3.1 Some Properties of the NLS Consider the NLS in the form Note: ∂tψ = i 2 ∂2 x ψ + is|ψ|2 ψ with s = ±1 • the magnitude of the coefficients can be absorbed into the amplitude and the spatial scale • the overall sign of the r.h.s. can be absorbed by running time backward t → −t • the relative sign s between ∂ 2 xψ and |ψ| 2 ψ cannot be changed by scaling or coordinate transformations – s = +1: focusing case (spatially homogeneous oscillations linearly unstable, cf. Benjamin-Feir instability of CGL). – s = −1: defocusing case. The NLS does not have a Lyapunov functional, but is a Hamiltonian system with Hamiltonian (energy) functional i.e. H{ψ, ψ ∗ } = 1 � 2 |∂xψ| 2 − |ψ| 4 dx ∂tψ = −i δH{ψ, ψ∗ ) δψ∗ since using integration by parts and employing the basic property of functional derivatives 4 , δψ(x) δψ(x ′ ) = δ(x − x′ ), one gets δH{ψ, ψ∗ ) δψ∗ Conserved Quantities: • L2-norm of ψ: N = � |ψ| 2 dx d N = dt = = δ δψ∗ � 1 2 = ���� integration by parts −∂ 2 x ψ ψ∗ − ψ 2 ψ ∗2 dx = − 1 2 ∂2 x ψ − ψ2 ψ ∗ � (34) ∂tψ ψ ∗ + ψ ∂tψ ∗ dx = (35) � � i 2 ∂2 x ψ + i|ψ|2 ψ � i 2 ∂2 x ψ ψ∗ − � i 2 ∂2 x ψ � ψ ∗ − ψ � ψ ∗ dx = 0 � i 2 ∂2 x ψ∗ + i|ψ| 2 ψ ∗ � dx 4ψ(x) can be thought of as a vector with x labeling its component. The functional derivative is then analogous to dvi dvj = δij for a vector v = (v1, . . . , vn). 77

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