Celebratio Mathematica

Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Cambridge, MA), 1998. MR 1726558 Zbl 0918.00013 book

C.-L. Terng and K. Uhlenbeck: “Poisson actions and scattering theory for integrable systems,” pp. 315–402 in Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Boston), 1998. MR 1726931 Zbl 0935.35163 ArXiv dg-ga/9707004 incollection

Conservation laws, hierarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS \( n{\times}n \) hierarchy (which includes the non-linear Schrödinger equation, modified KdV, and the \( n \)-wave equation). We first find a simple model Poisson group action that contains flows for systems with a Lax pair whose terms all decay on \( R \). Bäcklund transformations and flows arise from subgroups of this single Poisson group. For the ZS-AKNS \( n{\times}n \) hierarchy defined by a constant \( a\in\mathfrak{u}(n) \), the simple model is no longer correct. The \( a \) determines two separate Poisson structures. The flows come from the Poisson action of the centralizer \( H_a \) of \( a \) in the dual Poisson group (due to the behaviour of \( e^{a\lambda x} \) at infinity). When \( a \) has distinct eigenvalues, \( H_a \) is abelian and it acts symplectically. The phase space of these flows is the space \( S_a \) of left cosets of the centralizer of \( a \) in \( D_- \), where \( D_- \) is a certain loop group. The group \( D_- \) contains both a Poisson subgroup corresponding to the continuous scattering data, and a rational loop group corresponding to the discrete scattering data. The \( H_a \)-action is the right dressing action on \( S_a \). Bäcklund transformations arise from the action of the simple rational loops on \( S_a \) by right multiplication. Various geometric equations arise from appropriate choice of \( a \) and restrictions of the phase space and flows. In particular, we discuss applications to the sine-Gordon equation, harmonic maps, Schrödinger flows on symmetric spaces, Darboux orthogonal coordinates, and isometric immersions of one space-form in another.

C.-L. Terng and K. Uhlenbeck: “Introduction,” pp. 5–19 in Integral systems. Edited by C.-L. Terng and K. Uhlenbeck. Surveys in Differential Geometry 4. International Press (Cambridge, MA), 1998. Zbl 0938.35182 incollection

C.-L. Terng and K. Uhlenbeck: “Bäcklund transformations and loop group actions,” Comm. Pure Appl. Math. 53 : 1 (2000), pp. 1–75. MR 1715533 Zbl 1031.37064 article

We construct a local action of the group of rational maps from \( \mathbb{S}^2 \) to \( \mathrm{GL}(n,\mathbb{C}) \), on local solutions of flows of the ZS-AKNS \( \mathfrak{sl}(n,\mathbb{C}) \)-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different factorizations of a quadratic element. We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities. However, the action of the subgroup of the rational maps that satisfy the \( U(n) \)-reality condition \[ g(\overline{\lambda})*g(\lambda) = I \] on the space of global rapidly decaying solutions of the flows in the \( \mathfrak{u}(n) \)-hierarchy is global, and the action of a simple element gives a global Bäcklund transformation. The actions of certain elements in the rational loop group on the vacuum give rise to explicit time-periodic multisolitons (multibreathers). We show that this theory generalizes the classical Bäcklund theory of the sine-Gordon equation. The group structures of Bäcklund transformations for various hierarchies are determined by their reality conditions. We identify the reality conditions (the group structures) for the \( \mathfrak{sl}(n,\mathbb{R}) \), \( \mathfrak{u}(k,n-k) \), KdV, Kupershmidt–Wilson, and Gel’fand–Dikii hierarchies. The actions of linear fractional transformations that satisfy a reality condition, modulo the center of the group of rational maps, give Bäcklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under this reality condition, the action is again local, and Bäcklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed.

C.-L. Terng and K. Uhlenbeck: “Geometry of solitons,” Notices Am. Math. Soc. 47 : 1 (2000), pp. 17–25. cover article. MR 1733063 Zbl 0987.37072 article

C.-L. Terng and K. Uhlenbeck: “\( 1+1 \) wave maps into symmetric spaces,” Comm. Anal. Geom. 12 : 1–2 (2004), pp. 345–388. MR 2074882 Zbl 1082.37068 article

We explain how to apply techniques from integrable systems to construct \( 2k \)-soliton homoclinic wave maps from the periodic Minkowski space \( \mathbb{S}^1\times\mathbb{R}^1 \) to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the \( -1 \) flow equation associated to a compact Lie group \( G \) and wave maps into \( G \). We use Bäcklund transformations to construct explicit \( 2k \)-soliton breather solutions for the \( -1 \) flow equation and show that the corresponding wave maps are periodic and homoclinic. The compact symmetric space \( G/K \) can be embedded as a totally geodesic submanifold of \( G \) via the Cartan embedding. We prescribe the constraint condition for the \( -1 \) flow equation associated to \( G \) which insures that the corresponding wave map into \( G \) actually lies in \( G/K \). For example, when \[ G/K = \mathrm{SU}(2)/\mathrm{SO}(2) = \mathbb{S}^2 ,\] the constrained \( -1 \)-flow equation associated to \( \mathrm{SU}(2) \) has the sine-Gordon equation (SGE) as a subequation and classical breather solutions of the SGE are 2-soliton breathers. Thus our result generalizes the result of Shatah and Strauss that a classical breather solution of the SGE gives rise to a periodic homoclinic wave map to \( \mathbb{S}^2 \). When the group \( G \) is non-compact, the bi-invariant metric on \( G \) is pseudo-Riemannian and Bäcklund transformations of a smooth solution often are singular. We use Bäcklund transformations to show that there exist smooth initial data with constant boundary conditions and finite energy such that the Cauchy problem for wave maps from \( \mathbb{R}^{1,1} \) to the pseudo-Riemannian manifold \( \mathrm{SL}(2,\mathbb{R}) \) develops singularities in finite time.

C.-L. Terng and K. Uhlenbeck: “Schrödinger flows on Grassmannians,” pp. 235–256 in Integrable systems, geometry, and topology. Edited by C.-L. Terng. AMS/IP Studies in Advanced Mathematics 36. American Mathematical Society (Providence, RI), 2006. MR 2222517 Zbl 1110.37056 ArXiv math/9901086 incollection

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold \( M \) of \( k \)-planes in \( C^n \) is the evolution equation on the space \( C(\mathbb{R},M) \) of paths on \( M \): \[ J_{\lambda}(\gamma_{\lambda}) = \nabla_{\gamma_x}\gamma_x, \] where \( \nabla \) is the Levi–Civita connection of the Kähler metric and \( J \) is the complex structure. 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 that is gauge equivalent to the Lax pair of the matrix non-linear Schrödinger equation (MNLS) for \( q \) from \( \mathbb{R}^2 \) to the space of complex \( k{\times}(n-k) \) matrices: \[ q_t = \tfrac{i}{2}(q_{xx} + 2qq^*q). \] We construct via gauge transformations an isomorphism from \( C(\mathbb{R},M) \) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order \( k \) symplectic structures correspond to the order \( k-2 \) symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice.

B. Dai, C.-L. Terng, and K. Uhlenbeck: “On the space-time monopole equation,” pp. 1–30 in Essays in geometry in memory of S. S. Chern. Edited by S.-T. Yau. Surveys in Differential Geometry 10. International Press (Somerville, MA), 2006. MR 2408220 Zbl 1157.53016 incollection

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang–Mills equation on \( \mathbb{R}^{2,2} \). A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas–Ioannidou, Villarroel and Zakhorov–Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

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C.-L. Terng and K. Uhlenbeck: “The \( n{\times}n \) KdV hierarchy,” J. Fix. Point Theory A. 10 : 1 (2011), pp. 37–61. MR 2825739 Zbl 1251.37070 article

C.-L. Terng and K. Uhlenbeck: “Tau function and Virasoro action for the \( n{\times}n \) KdV hierarchy,” Comm. Math. Phys. 342 : 1 (2016), pp. 81–116. MR 3455146 Zbl 1354.37068 article

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group \( L \) as positive and negative subgroups \( L_{\pm} \) and a commuting sequence in the Lie algebra \( \mathcal{L}_+ \) of \( L_+ \). Given \( f\in L_- \), there is a formal inverse scattering solution \( u_f \) of the hierarchy. When there is a 2 co-cycle on \( \mathcal{L} \) that vanishes on both \( \mathcal{L}_+ \) and \( \mathcal{L}_- \), Wilson constructed for each \( f\in L_- \) a tau function \( \tau_f \) for the hierarchy. In this third paper, we prove the following results for the \( n{\times}n \) KdV hierarchy:

The second partials of \( \ln\tau_f \) are differential polynomials of the formal inverse scattering solution \( u_f \). Moreover, \( u_f \) can be recovered from the second partials of \( \ln\tau_f \).
The natural Virasoro action on \( \ln\tau_f \) constructed in the second paper is given by partial differential operators in \( \ln\tau_f \).
There is a bijection between phase spaces of \( n{\times}n \) KdV hierarchy and Gelfand–Dickey (\( \mathrm{GD}_n \)) hierarchy on the space of order \( n \) linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection.
Our Virasoro action on the \( n{\times}n \) KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the \( \mathrm{GD}_n \) hierarchy under the bijection.

C.-L. Terng and K. Uhlenbeck: “Tau functions and Virasoro actions for soliton hierarchies,” Comm. Math. Phys. 342 : 1 (2016), pp. 117–150. MR 3455147 Zbl 1346.37058 article

There is a general method for constructing a soliton hierarchy from a splitting \( L_{\pm} \) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \( \mathcal{L}_+ \). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each \( f \) in the negative subgroup \( L_- \) a solution \( u_f \) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \( \tau_f \) for each element \( f\in L_- \). In this paper, we give integral formulas for variations of \( \ln \tau_f \) and second partials of \( \ln \tau_f \), discuss whether we can recover solutions \( u_f \) from \( \tau_f \), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \( \mathrm{GL}(n,\mathbb{C}) \)-hierarchy.

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