Schrödinger flows on Grassmannians. (English) Zbl 1110.37056

Terng, Chuulian; Uhlenbeck, Karen

Terng, Chuu-Lian (ed.), Integrable systems, geometry, and topology. Providence, RI: American Mathematical Society (AMS). Somerville, MA: International Press. (ISBN 0-8218-4048-7/pbk). AMS/IP Studies in Advanced Mathematics 36, 235-256 (2006).

The authors of this very interesting paper study the so-called geometric nonlinear Schrödinger equation (GNLS) on the complex Grassmannian manifold \(M\) (target) of \(k\)-planes in \(\mathbb{C}^n\)
\[ J_{\gamma }(\gamma_t)=\nabla_{\gamma_x}\gamma_x , \]
where \(\nabla \) is the Levi-Civita connection of the Kähler metric \(g\) and \(J\) is a complex structure. It is an evolution equation on the space \(C(\mathbb R,M)\) of paths on \(M\). GNLS is the Hamiltonian equation for the energy functional on \(C(\mathbb R,M)\) with respect to the symplectic form induced from the Kähler form on \(M\). It has a Lax pair of the matrix nonlinear Schrödinger equation (MNLS) for \(q\) from \(\mathbb R^2\) to the space of complex \(k\times (n-k)\) matrices
\[ q_t=(i/2)(q_{xx}+2qq^{\ast }q), \]
where \(q\) is a map from \(\mathbb{R}^2\) to the space \(\mathcal{M}_{k\times (n-k)}\) of complex \(k\times (n-k)\)-matrices. This equation is studied by A. P. Fordy and B. P. Kulish [Commun. Math. Phys. 89, 427–443 (1983; Zbl 0563.35062)]. An isomorphism from \(C(\mathbb R,M)\) to the phase space of the MNLS equation is defined. Thus the GNLS corresponds to the MNLS flow. It is shown that the space of conservation laws has a structure of a non-abelian Poisson group. A hierarchy of symplectic structures for GNLS is determined. A sequence of symplectic structures of order \(k\) on the phase space of the GNLS is constructed. It is shown that they are the pull back of the known order \(k+2\) symplectic form on the phase space of the MNLS equation under some isomorphism from the phase space of GNLS to the phase space of MNLS. Two standard symplectic forms for the KdV equation are constructed as well.
For the entire collection see [Zbl 1086.14002].

Reviewer: Dimitar A. Kolev (Sofia)

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58J99	Partial differential equations on manifolds; differential operators
37K25	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

Schrödinger flow; Grassmannian; symplectic structure; Hamiltonian equation; Euler-Lagrange equation; Riemannian manifold

Citations:

Zbl 0563.35062