Ivancevic_Applied-Diff-Geom

382 Applied Differential Geometry: A Modern Introductionalgebra χ(P) of observables on P consisting of real–analytic functionalderivatives δF /δψ, δF /δ ¯ψ ∈ S(R, C).The Hamiltonian function H : P −→ R is given by∫ (+∞∣∣∣∣ )2∂ψH(ψ) =∂x ∣ + χ|ψ| 4 dx−∞and is equal to the total energy of the soliton. It is a conserved quantityfor (4.3) (see [Seiler (1995)]).The Poisson bracket on χ(P) represents a direct generalization of theclassical nD Poisson bracket{F, G} + (ψ) = i∫ +∞−∞( δFδψδG δF−δ ¯ψ δ ¯ψ)δGdx. (3.192)δψIt manifestly exhibits skew–symmetry and satisfies Jacobi identity. Thefunctionals are given by δF /δψ = −i{F, ¯ψ} and δF /δ ¯ψ = i{F, ψ}. Thereforethe algebra of observables χ(P) represents the Lie algebra and thePoisson bracket is the (+) Lie–Poisson bracket {F, G} + (ψ).The nonlinear Schrödinger equation (3.191) for the solitary particle–wave is a Hamiltonian system on the Lie algebra χ(P) relative to the (+)Lie–Poisson bracket {F, G} + (ψ) and Hamiltonian function H(ψ). Thereforethe Poisson manifold (χ(P), {F, G} + (ψ)) is defined and the abstractPoisson evolution equation (3.179), which holds for any smooth functionF : χ(P) →R, is equivalent to equation (3.191).A more subtle model of soliton dynamics is provided by the Korteveg–deVries equation [Ivancevic and Pearce (2001a)]f t − 6ff x + f xxx = 0, (f x = ∂ x f), (3.193)where x ∈ R and f is a real–valued smooth function defined on R (comparewith (3.81) above). This equation is related to the ordinary Schrödingerequation by the inverse scattering method [Seiler (1995); Ivancevic andPearce (2001a)].We may define the infinite–dimensional phase–space manifold V = {f ∈S(R)}, where S(R) is the Schwartz space of rapidly–decreasing real–valuedfunctions R). We define further χ(V) to be the algebra of observablesconsisting of functional derivatives δF /δf ∈ S(R).The Hamiltonian H : V → R is given byH(f) =∫ +∞−∞(f 3 + 1 2 f 2 x) dx