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The KdV curve and Schrödinger-Airy curve. (English) Zbl 1248.37067

Let \((N,\omega,J)\) be a Kähler manifold and consider the functionals \[ F(u)=\frac{1}{2}\int_{S^1}\langle\nabla_x u_x, Ju_x\rangle dx \] and \[ E(u)=\frac{1}{2}\int_{S^1}|u_x|^2dx, \] defined for \(u\in W^{2,2}(S^1,N)\). A KdV curve is a critical point of \(F\), a Schrödinger-Airy curve a critical point of \(F\) under the constraint \(E(u)\equiv \mathrm{const}\). The paper presents examples and basic properties of these curves. The author also proves an existence result for critical points of \(F_H(u)=F(u)+\int_{S^1}H(x,u)dx\) when \(N=T^{2n}\) is the flat torus and \(H:S^1\times T^{2n}\to \mathbb{R}\) a Hamiltonian.

MSC:

37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
53C99 Global differential geometry
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References:

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