arXiv subjects

Chuu-Lian Terng

Publications and source records attributed to Chuu-Lian Terng.

30 records · Page 2Linked to original sources

arXiv2006

On the space-time Monopole equation

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on $\R^{2,2}$. 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.

math.DG↗

arXiv2006

Associative Cones and Integrable Systems

We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit 6-sphere. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S^6 is the equation for primitive maps associated to the 6-symmetric space G_2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice associated to G_2, and a discussion of periodic solutions is given.

math.DG↗

arXiv2004

Curved flats, exterior differential systems, and conservation laws

Let $σ$ be an involution of a real semi-simple Lie group $U$, $U_0$ the subgroup fixed by $σ$, and $U/U_0$ the corresponding symmetric space. Ferus and Pedit called a submanifold $M$ of a rank $r$ symmetric space $U/U_0$ a {\it curved flat} if $T_pM$ is tangent to an $r$-dimensional flat of $U/U_0$ at $p$ for each $p\in M$. They noted that the equation for curved flats is an integrable system. Bryant used the involution $σ$ to construct an involutive exterior differential system $\ci_σ$ such that integral submanifolds of $\ci_σ$ are curved flats. Terng used $r$ first flows in the $U/U_0$-hierarchy of commuting soliton equations to construct the $U/U_0$-system. She showed that the $U/U_0$-system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the Gauss-Codazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system $\ci_σ$.

math.DG↗

arXiv2004

Backlund transformations, Ward solitons, and unitons

The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or triple pole. Some of the main results of this paper are: (i) We construct algebraic Bäcklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order $k$ limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions in $x, y$ and $t$. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all $k$-unitons from finitely many rational maps from $C$ to $C^n$.

math.DG↗

arXiv2004

Periodic and homoclinic solutions of the modified 2+1 Chiral model

We use algebraic Backlund transformations (BTs) to construct explicit solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where $T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and holomorphic maps $π$ from $T^2$ to Gr$(k,C^n)$. We apply Bäcklund transformations with carefully chosen poles and $π$'s to construct infinitely many solutions of the 2+1 chiral model that are (i) doubly periodic in space variables and periodic in time, i.e., triply periodic, (ii) homoclinic in the sense that the solution $u$ has the same stationary limit $u_0$ as $t\to \pm\infty$ and is tangent to a stable linear mode of $u_0$ as $t\to\infty$ and is tangent to an unstable mode of $u_0$ as $t\to -\infty$.

math.DG↗

arXiv2003

1+1 wave maps into symmetric spaces

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ 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 $G/K=SU(2)/SO(2)=S^2$, the constrained -1-flow equation associated to 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 $S^2$. 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 $R^{1,1}$ to the pseudo-Riemannian manifold $SL(2,R)$ develops singularities in finite time.

math.DG↗

arXiv2002

Geometries and Symmetries of Soliton equations and Integrable Elliptic equations

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation for harmonic maps from the plane to a compact Lie group, for primitive maps from the plane to a $k$-symmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of (i) construction of solutions using loop group factorizations, (ii) PDEs in differential geometry that are soliton equations or elliptic integrable systems, (iii) similarities and differences of soliton equations and integrable elliptic systems.

math.DG↗

arXiv2001

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint $U$-orbits from flows in the soliton hierarchy associated to a compact Lie group $U$. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.

math.DG↗

arXiv2000

The submanifold geometries associated to Grassmannian systems

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-dimensional linear subspaces in R^{m+n}, and G_{m,n}^1 the Grassmannian of space like m-dimensional linear subspaces in the Lorentzian space R^{m+n,1}. In this paper, we use techniques from soliton theory to study submanifolds in space forms whose Gauss-Codazzi equations are gauge equivalent to the G_{m,n}-system or the G_{m,n}^1-system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G_{m,n} and G_{m,n}^1 systems correspond to Bäcklund, Darboux and Ribaucour transformations for submanifolds.

math.DG↗

arXiv1999

Schrodinger flows on Grassmannians

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations an isomorphism from C(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.

math.DG↗

arXiv1998

Bäcklund Transformations and Loop Group Actions

We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-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 $g(\barł)^*g(ł)=I$ on the space of global rapidly decaying solutions of the flows in the $u(n)$-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 multi-solitons (multi-breathers). 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 $sl(n,R)$, $u(k,n-k)$, KdV, Kupershmidt-Wilson, and Gel'fand-Dikii hierarchies. The actions of linear fractional transformations that satisfies a reality condition, modulo the center of the group of rational maps, gives Bäcklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under these reality condition, the action is again local, and Bäcklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed.

math.DG↗

arXiv1997

Poisson Actions and Scattering Theory for Integrable Systems

Conservation laws, heirarchies, 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 nxn heirarchy (which includes the non-linear Schrödinger equation, modified KdV, and the n-wave equation). We also discuss a number of applications in geometry, including the sine-Gordon equation, harmonic maps, Schrödinger flows on Hermitian symmetric spaces, Darboux orthogonal coordinates, and isometric immerisons of one space form in another.

dg-ga↗