The Toda Flow as a Porous Medium Equation

Abstract

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

Appendices

Appendix A: Universal Double Bracket Flows

1.1 The gradient flow on a Lie group

Here we consider a framework for Riemannian gradient flows confined to group orbits. This leads to the proper setting for the double bracket flows in the case of an arbitrary inertia operator. A similar framework, with focus on shape analysis, is also given in [1].

Let Q be a configuration manifold, possibly infinite-dimensional. Furthermore, let G be a Lie group (or a Fréchet–Lie group in the infinite-dimensional case) acting on Q from the left by a smooth action map $\Phi :G\times Q\rightarrow Q$; the action of $g\in G$ on $q\in Q$ is denoted $g\cdot q$. This action is typically neither free, nor transitive. The orbit of $q\in Q$ is

$$\begin{aligned} \textrm{Orb}(q) = \{ g\cdot q\mid q\in G \}. \end{aligned}$$

We think of Q as the space of ‘shapes’ and the orbit $\textrm{Orb}(q)$ represents all possible deformations of q. For a given template shape $q_0$, our objective is to study gradient flows on $\textrm{Orb}(q_0)$. Notice, however, that we do not assume that Q is Riemannian; the Riemannian structure on $\textrm{Orb}(q_0)$ is instead inherited from G. Before we state the main result of this section, we need a few concepts from geometric mechanics (cf. [17]).

By differentiating the action map at the identity we obtain the infinitesimal action map ${\mathfrak {g}}\times Q\rightarrow T Q$, where ${\mathfrak {g}}= T_e G$ is the Lie algebra of G. The infinitesimal action of $v\in {\mathfrak {g}}$ on $q\in Q$ is denoted $v\cdot q$. This linear map in v is the (cotangent bundle) momentum map:

Definition 5.1

The momentum map $J:T^*Q\rightarrow {\mathfrak {g}}^*$ is defined by

$$\begin{aligned} \left\langle J(q,p),v \right\rangle = \left\langle p,v\cdot q \right\rangle \qquad \forall \, v\in {\mathfrak {g}}, \end{aligned}$$

where $T^*Q$ denotes the cotangent bundle of Q.¹

Next, we introduce a Riemannian structure on G.

Definition 5.2

A Riemannian metric $\langle \!\langle \cdot ,\cdot \rangle \!\rangle :TG\times TG\rightarrow {{\mathbb {R}}}$ on G is called right-invariant if

$$\begin{aligned} \langle \!\langle u,v \rangle \!\rangle _e = \langle \!\langle u\cdot g, v\cdot g \rangle \!\rangle _{g}, \qquad \forall \, g\in G,\quad \forall \, u,v\in {\mathfrak {g}}, \end{aligned}$$

where $u\cdot g$ denotes the tangent lifted right action of g on u.

A right-invariant metric is completely determined by the inner product $\langle \!\langle \cdot ,\cdot \rangle \!\rangle _e$.

Definition 5.3

Let $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$ be a right-invariant metric on G. Then the inertia operator $A:{\mathfrak {g}}\rightarrow {\mathfrak {g}}^*$ is defined by

$$\begin{aligned} \left\langle Av,u \right\rangle = \langle \!\langle v,u \rangle \!\rangle _e. \end{aligned}$$

Since G acts on Q from the left, the action map induces a Riemannian structure on $\textrm{Orb}(q)$. To see this, we first need the notion of horizontal vectors on G.

Definition 5.4

The vertical distribution associated with the action of G on $q\in Q$ is the subbundle of TG given by

$$\begin{aligned} \textrm{Ver}_g = \{ v\cdot g \in T_g G \mid v\cdot g\cdot q = 0 \}. \end{aligned}$$

If $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$ is a right-invariant metric on G, then the corresponding horizontal distribution is given by

$$\begin{aligned} \textrm{Hor}_g = \textrm{Ver}_g^\bot \end{aligned}$$

where the complement is taken with respect to $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$.

Lemma 5.5

Assume that $\textrm{Orb}(q)$ is a submanifold of Q. Then any right-invariant Riemannian metric $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$ induces a Riemannian metric ${\textsf{g}}$ on $\textrm{Orb}(q)$ fulfilling

$$\begin{aligned} {\textsf{g}}_{g\cdot q}(v\cdot g\cdot q,v\cdot g\cdot q) = \langle \!\langle v,v \rangle \!\rangle _e \qquad \forall \, v\cdot g\in \textrm{Hor}_g. \end{aligned}$$

(5.1)

Proof

Since $\textrm{Orb}(q)$ is a manifold, the mapping $\pi :G\rightarrow Q$ defined by $\pi (g) = g\cdot q$ is a submersion. Thus, $T_g \pi :\textrm{Hor}_g \rightarrow T_{\pi (g)}\textrm{Orb}(q)$ is a linear isomorphism. Now, for $(x,\dot{x}) \in T\textrm{Orb}(q)$, take any g such that $x = \pi (g)$ (which exist by definition of the group orbit). Define the metric at x by

$$\begin{aligned} {\textsf{g}}_x(\dot{x},\dot{x}) = \langle \!\langle \underbrace{(T_g\pi )^{-1}\dot{x}}_{v\cdot g},\underbrace{(T_g\pi )^{-1}\dot{x}}_{v\cdot g} \rangle \!\rangle _g. \end{aligned}$$

By right-invariance, this metric satisfies (5.1) and is independent of the choice of g. $\square$

Let G be a Lie group acting from the left on a manifold Q of shapes. Let $q_0\in Q$ and let $F:Q\rightarrow {{\mathbb {R}}}$ be a function on Q. We are interested in finding the minimum of F on the G-orbit of $q_0$, that is, we want to minimize the function $E:G\rightarrow {{\mathbb {R}}}$ defined by

$$\begin{aligned} E(g) = F(g\cdot q_0). \end{aligned}$$

If G is equipped with a right-invariant Riemannian metric ${\textsf{g}}$, defined by an inertia operator $A:{\mathfrak {g}}\rightarrow {\mathfrak {g}}^*$, one can define the gradient vector field $\nabla E$ on G by

$$\begin{aligned} {\textsf{g}}_g(\nabla E(g),\dot{g}) = \left\langle \textrm{d}E,\dot{g} \right\rangle . \end{aligned}$$

Our aim is to solve the minimization problem by considering the gradient flow

$$\begin{aligned} \dot{g} = -\nabla E(g). \end{aligned}$$

(5.2)

Proposition 5.6

The gradient $\nabla E$ is given by

$$\begin{aligned} \nabla E(g) = \xi \cdot g. \end{aligned}$$

where $\xi \in {\mathfrak {g}}$ is given by

$$\begin{aligned} \xi = A^{-1}J(g\cdot q_0, \textrm{d}F(g\cdot q_0)). \end{aligned}$$

Proof

By definition of the gradient we have

$$\begin{aligned} \langle \!\langle \nabla E, \dot{g} \rangle \!\rangle _{g} = \frac{d}{dt}E(g(t)) = \left\langle d F,\xi \cdot (g\cdot q_0) \right\rangle . \end{aligned}$$

where $\xi = \dot{g}\cdot g^{-1}$. Now from the definition of the momentum map it follows that

$$\begin{aligned} \langle \!\langle \nabla E, \dot{g} \rangle \!\rangle _{g} = \left\langle J(g\cdot q_0, dF), \xi \right\rangle = \langle \!\langle A^{-1}J(g\cdot q_0, dF), \dot{g}\cdot g^{-1} \rangle \!\rangle _e \end{aligned}$$

The result follows since the metric is right invariant. $\square$

Proposition 5.7

The gradient flow (5.2) induces a gradient flow on the G-orbit of $q_0$, given by

$$\begin{aligned} \dot{q} = -u(q)\cdot q \end{aligned}$$

(5.3)

where

$$\begin{aligned} u(q) = A^{-1}J(q, \textrm{d}F(q)) \end{aligned}$$

Proof

Follows from Prop. 5.6. $\square$

Definition 5.8

The double-bracket flow on the dual Lie algebra ${\mathfrak {g}}^*$ with the inertia operator A and a potential function F on ${\mathfrak {g}}^*$ is as follows:

$$\begin{aligned} \dot{m} = {\text {ad}}^*_{A^{-1}{\text {ad}}^*_{\textrm{d}F(m)}(m)}(m). \end{aligned}$$

For the quadratic potential $F(m) = \langle m, A^{-1}m\rangle$ the corresponding flow is

$$\begin{aligned} \dot{m} = {\text {ad}}^*_{A^{-1}{\text {ad}}^*_{A^{-1}m}(m)}(m). \end{aligned}$$

Proposition 5.9

The double-bracket flow on ${\mathfrak {g}}^*$ is the gradient for the restriction of F on each coadjoint orbit.

Proof

Indeed, consider the special case where $Q = {\mathfrak {g}}^*$. The action is given by $g\cdot m = {\text {Ad}}_g^* m$. The momentum map is thereby given by

$$\begin{aligned} \langle J(m,\xi ), v \rangle = \langle \xi , {\text {ad}}^*_v(m)\rangle = \langle {\text {ad}}_v(\xi ), m\rangle = \langle -{\text {ad}}_\xi (v), m\rangle = \langle -{\text {ad}}^*_\xi (m), v\rangle \end{aligned}$$

Then from (5.3) we obtain the equation(s) in Definition 5.8, and the statement follows from Proposition 5.7. The corresponding flow tries to minimize the energy on a specific co-adjoint orbit. $\square$

Note also that this flow is always orthogonal (w.r.t. $A^{-1}$) to the (Hamiltonian) Euler-Arnold flow: one of them is tangent to levels of the Hamiltonian, while the other is orthogonal to the levels of the same function regarded as a potential. One can also see this directly, since

$$\begin{aligned}{} & {} \langle {\text {ad}}^*_{A^{-1}{\text {ad}}^*_{A^{-1}m}(m)}(m), A^{-1}{\text {ad}}^*_{A^{-1}m}(m)\rangle \\{} & {} \quad =\langle m, [A^{-1}{\text {ad}}^*_{A^{-1}m}(m), A^{-1}{\text {ad}}^*_{A^{-1}m}(m)] \rangle = 0. \end{aligned}$$

1.2 Contraction property of the flow

Consider the gradient flow (5.2).

Proposition 5.10

Let $d:G\times G\rightarrow {{\mathbb {R}}}$ denote the Riemannian distance function induced by the right-invariant metric associated with the gradient flow (5.2). Let $\gamma :[0,T)\rightarrow G$ be a solution curve. Then, for all $t\in [0,T)$

$$\begin{aligned} d(\gamma (0),\gamma (t))^2 \le t\Big ( E\big (\gamma (0)\big ) - E\big (\gamma (t)\big ) \Big ). \end{aligned}$$

Proof

First notice that

$$\begin{aligned} \frac{d}{dt} E(\gamma (t)) = \langle \!\langle \nabla E(\gamma ),{\dot{\gamma }} \rangle \!\rangle _\gamma = -\langle \!\langle {\dot{\gamma }},{\dot{\gamma }} \rangle \!\rangle _\gamma = -\langle \!\langle {\dot{\gamma }}\circ \gamma ^{-1}, {\dot{\gamma }}\circ \gamma ^{-1} \rangle \!\rangle _e = -\langle \!\langle v,v \rangle \!\rangle _e \end{aligned}$$

for the vector field v(t) generating the curve $\gamma (t)$. Denote by $\Vert v\Vert ^2_A:= \langle \!\langle v,v \rangle \!\rangle _e = \left\langle Av,v \right\rangle$. Since any curve between $\gamma (0)$ and $\gamma (t)$ cannot exceed the length of the geodesic between the points we have

$$\begin{aligned} d(\gamma (0),\gamma (t)) \le \int _0^t \Vert v(s)\Vert _A \,\textrm{d}s \le \sqrt{t}\left( \int _0^t \Vert v(s)\Vert _A^2\,\textrm{d}s \right) ^{1/2}, \end{aligned}$$

where the last inequality is Cauchy–Schwartz on $L^2([0,t])$. $\square$

Remark 5.11

Theorem 5.10 in relation to the IPM flow then gives the following result: the “Arnold fluid-distance” between $\varphi (0)$ and $\varphi (t)$ is bounded by

$$\begin{aligned} d(\varphi (0),\varphi (t))^2 \le t \int _M V(\rho (0)-\rho (t)). \end{aligned}$$

Appendix B: A Direct Continuous Toda Limit

Start from the Hamiltonian for the finite-dimensional Toda lattice in $(q_j,p_j)$ variables. Take the limit $n\rightarrow \infty$ to obtain a continuous system on $T^*\textrm{Dens}({\mathbb {R}})$. Under this limit we have

$$\begin{aligned} q_{j+1}-q_{j} \rightarrow \varphi '(z) \end{aligned}$$

where $\varphi$ is the diffeomorphism describing how each point on the line has moved. Since we also have $\varphi '(z) > 0$ and therefore $\rho = \textrm{Jac}(\varphi ^{-1}) = 1/\varphi '\circ \varphi$, we get the potential

$$\begin{aligned} U(\varphi ) = \int _{\mathbb {R}} V(\varphi ') dz = \int _{\mathbb {R}} V(1/\rho \circ \varphi ^{-1})dz, \end{aligned}$$

where the potential V is

$$\begin{aligned} V(r) =\exp (-2r). \end{aligned}$$

The gradient of U is computed as

$$\begin{aligned} \frac{d}{dt}U(\varphi ) = \int _{\mathbb {R}}V'(\varphi '){\dot{\varphi }}' dz = \int _{\mathbb {R}} \left( -\frac{\partial }{\partial x}V'(\varphi ')\right) {\dot{\varphi }}\; dz \end{aligned}$$

Newton’s equations on the space of diffeomorphisms is thereby

$$\begin{aligned} {\ddot{\varphi }} = \frac{\partial }{\partial z}V'(\varphi ') = -2\frac{\partial }{\partial z} \exp (-2\varphi '). \end{aligned}$$

The fluid formulation of this system is

$$\begin{aligned} \dot{v} + \nabla _v v = \left( \frac{\partial }{\partial z}V'(\varphi ') \right) \circ \varphi ^{-1}, \qquad {\dot{\varphi }} = v\circ \varphi . \end{aligned}$$

Let us now make the following change of variables

$$\begin{aligned} a = \exp (-\varphi '), \qquad b = {\dot{\varphi }}. \end{aligned}$$

(6.1)

Since the density is uniform, the variable b is interpreted physically as the Lagrangian momentum. Direct calculations now yield

$$\begin{aligned} \dot{a} = -a \frac{\partial }{\partial z}{\dot{\varphi }} = -a \frac{\partial }{\partial z}b \end{aligned}$$

and, likewise,

$$\begin{aligned} \dot{b} = {\ddot{\varphi }} = -2\frac{\partial }{\partial z}a^2. \end{aligned}$$

This is the same equation as in the first continuous limit, which can be rewritten as

$$\begin{aligned} \dot{a} = -a \frac{\partial }{\partial z}b, \qquad \dot{b} = -2 \frac{\partial }{\partial z}a^2. \end{aligned}$$

(6.2)