Whitham equations

Perhaps the proper beginning of Whitham theory is Whitham's work [a75], [a74], which can be viewed as a crucible of various averaging ideas subsequently developed in e.g. [a3], [a4], [a12], [a13], [a14], [a27], [a28], [a29], [a42], [a49], [a50], [a61], [a73] to theories involving multi-phase averaging, Hamiltonian systems and weakly deformed soliton lattices. The term "Whitham equations" then became associated with the moduli dynamics of Riemann surfaces and this fits naturally into work on topological field theories, Frobenius manifolds, renormalization groups, coupling constants, and Seiberg–Witten theory (cf. also Seiberg–Witten equations), along with singularity theory, isomonodromy deformations, quantum cohomology and $K$-theory, Gromov–Witten invariants, Witten–Dijkgraaf–Verlinde–Verlinde equations, etc. (see the references below or the survey material in [a5], [a6], [a7], [a8], [a9], [a10]).

Averaging.[edit]

One of the most important applications of averaging theory and the Whitham equation is to the Korteweg–de Vries equation

$$\begin{array}{}\text{(a1)}& u_t-6u{u}_x+u_{xxx}=0.\end{array}$$

Key early papers on averaging for this equation include [a27] and [a49]. The basic ideas of G. Whitham are discussed in [a74], [a75]. Other important papers include [a1], [a2], [a25], [a65].

The key idea is averaging out fast scales; one introduces two scales: the "fast" scale $(x,t)$ and "slow" scale ($X=ϵx$, $T=ϵt$), small. One obtains a class of ( "finite-gap" ) solutions of the form

$$\begin{array}{}\text{(a2)}& u(x,t)=U=f_g({\theta }_1,\dots ,{\theta }_g),\end{array}$$

where $f_g$ is a meromorphic function of $g$ variables and ${\theta }_i={\kappa }_i+{\omega }_i+{\hat{\theta }}_i$, where the parameters $U$, ${\kappa }_i$, ${\omega }_i$ depend only on the slow variables.

One can then write down the evolution equations for $g$-phase wave trains in terms of differentials on an associated Riemann surface.

The Whitham equation for the Korteweg–de Vries equation is given by

$$\begin{array}{}\text{(a3)}& \frac{\partial d{\omega }_1}{\partial T}=\frac{\partial d{\omega }_3}{\partial X},\end{array}$$

where and are Abelian differentials on the Riemann surface of genus $g$ given by $y^2=R_g(\lambda )$ (cf. also Differential on a Riemann surface; Abelian differential; Riemann surface), where

$$R_g(\lambda )=\prod _{i=0}^{2g}(\lambda -{\lambda }_i)$$

and the branch points ${\lambda }_i$ are real and are assumed to satisfy ${\lambda }_0<\dots <{\lambda }_{2g}$.

Explicitly,

$$\begin{array}{}\text{(a4)}& d{\omega }_1(\lambda )=\frac{\prod _{i=1}^g(\lambda -{\alpha }_i)}{\sqrt{R_g(\lambda )}}d\lambda \sim \end{array}$$

$$\sim \frac{d\lambda }{\sqrt{\lambda }}+(\text{ holomorphic }),\text{ as }\lambda \to \mathrm{\infty },$$

$$\begin{array}{}\text{(a5)}& d{\omega }_3(\lambda )=\frac{{\lambda }^{g+1}-\frac{1}{2}{\sigma }_1{\lambda }^g+{\beta }_1{\lambda }^{g-1}+\dots +{\beta }_g}{\sqrt{R_g(\lambda )}}d\lambda \sim \end{array}$$

$$\sim \sqrt{\lambda }d\lambda +\text{ (holomorphic), as }\lambda \to \mathrm{\infty }.$$

Here, the coefficients $\{{\alpha }_j,{\beta }_j\}$ are determined by

$${\oint }_{A_j}d{\omega }_1={\oint }_{A_j}d{\omega }_3=0,j=1,\dots ,g,$$

the vanishing of the contour integral along the canonical $A_j$-cycle, and ${\sigma }_1=\sum _{i=0}^{2g}{\lambda }_i$.

Then the averaged solution of the Korteweg–de Vries equation is given by

$$\bar{u}(x,t)=\frac{1}{2}\sum _{i=0}^{2g}{\lambda }_i-\sum _{j=0}^g{\alpha }_j.$$

Note that when $g=0$, the equation reduces to the dispersionless Korteweg–de Vries equation (Hopf–Burgers equation) with ${\lambda }_0=2\bar{u}$ and ${\lambda }_1=\dots ={\lambda }_{2g}={\alpha }_1=\dots ={\alpha }_g=0$, i.e.,

$$\frac{\partial \bar{u}}{\partial T}=\bar{u}\frac{\partial \bar{u}}{\partial X}.$$

The Whitham equation for the discrete Toda lattice (cf. Toda lattices) is treated in [a4] where shock formation is analyzed. Shocks for the Korteweg–de Vries equation are analyzed in [a34], [a35].

The discrete Ablowitz–Ladik equations are analyzed in [a60].

The Whitham equations are also important in the analysis of the non-linear Schrödinger equation (cf. also Benjamin–Feir instability) and non-linear optics, see for example [a36], [a48], [a64] and references therein.

General Whitham theory.[edit]

More generally, for any compact Riemann surface ${\mathrm{\Sigma }}_g$ of genus $g$ and point $Q\sim \mathrm{\infty }$, the Baker–Akhiezer function $\psi$ gives rise to a KP-hierarchy (cf. also KP-equation). In particular, following [a11], [a27], [a42], $\psi$ can be written as

$$\begin{array}{}\text{(a6)}& \psi (P)=\exp (\sum t_n{\mathrm{\Omega }}_n)\varphi (\sum t_n{\stackrel{\rightharpoonup }{V}}_n,P),\end{array}$$

where $d{\mathrm{\Omega }}_n\sim d({\lambda }^n)+\dots$ near $\mathrm{\infty }$ and ${\int }_{A_i}d{\mathrm{\Omega }}_n=0$ with ${\int }_{B_i}d{\mathrm{\Omega }}_n=V_{in}\sim (\stackrel{\rightharpoonup }{V_n}{)}_i$. Here, $\varphi$ is periodic and ${\mathrm{\Omega }}_n={\mathrm{\Omega }}_n(T_m)$ with ${\stackrel{\rightharpoonup }{V}}_n={\stackrel{\rightharpoonup }{V}}_n(T_m)$ for slow times $T_m$ defined via $T_m=ϵt_m$ and $\stackrel{\rightharpoonup }{\theta }=\sum t_n{\stackrel{\rightharpoonup }{V}}_n$ is a point in the Jacobian $\mathrm{Jac}({\mathrm{\Sigma }}_g)$ (cf. also Jacobi variety). Assuming, for simplicity, that the periods are incommensurable, by ergodicity one finds

$$\frac{1}{2L}{\int }_{-L}^L\varphi d{t}_i=⟨\varphi ⟩={\left(\frac{1}{2\pi }\right)}^{2g}\int \dots \int \varphi d^{2g}\theta .$$

With ${\psi }^{\ast }$ corresponding to the adjoint Baker–Akhiezer function, one can think now of multi-scale analysis of $\psi {\psi }^{\ast }d\tilde{\mathrm{\Omega }}$ with ${\partial }_i\to {\partial }_i+ϵ(\partial /\partial T_i)$ plus averaging over the fast times (here, $d\tilde{\mathrm{\Omega }}=d\lambda +O({\lambda }^{-2})d\lambda$ near $\mathrm{\infty }$ is canonically specified). This corresponds to looking at an expansion and setting the average first-order term to zero, leading to the Whitham equations

$$\begin{array}{}\text{(a7)}& \frac{{\mathrm{\Omega }}_n}{\partial T_m}=\frac{\partial {\mathrm{\Omega }}_m}{\partial T_n}.\end{array}$$

Seiberg–Witten theory.[edit]

Given a low-energy effective action for an $N=2$ susy gauge theory with partition function

$$\begin{array}{}\text{(a8)}& Z(t,\varphi )={\int }_{{\varphi }_0}\mathcal{D}\varphi \exp [S(t,\varphi )],\end{array}$$

with fields, $t\sim$ coupling constants and gauge group in the background, it turns out (e.g. in matrix models) that $Z(t,\varphi )$ will often be a tau-function of KP–Toda type via Ward identities and Virasoro (origin of integrability). Recall that tau-functions are basic ingredients in integrable system theory (cf. also KP-equation; Toda lattices) and e.g.

$$\begin{array}{}\text{(a9)}& \psi =\frac{\exp (\sum t_n{\lambda }^n)\tau (t_j-(1/j{\lambda }^j))}{\tau (t_j)}.\end{array}$$

For $Z\sim \tau$ one has an effective (classical-type) dynamics in the $t$ variables and averaging corresponds in some sense to suppressing fast oscillations (which suggests a renormalization procedure); alternatively, it is also in some sense related to a quantization procedure in the first WKBJ approximation, which produces slow dynamics on the action variables (Hamiltonians $\sim$ Casimirs from $\tilde{g}=\text{ Lie }(G)$; cf. also Casimir element; Kac–Moody algebra), which is equivalent in many situations to dynamics on the moduli of the underlying spectral curves. Thus, the quantum arena shifts to the quantum moduli space and the $T_n$ appear as renormalized coupling constants in one approach and as deformation parameters of moduli in another. The tau-function $\tau$ goes to a quasi-classical tau-function whose logarithm (after adjustment) is called the pre-potential $F$ and this serves as a generating function for correlators and as a vehicle for expressing further renormalization effects. Consider (cf. [a31], [a32], [a33], [a37], [a38], [a39], [a40], [a41], [a62]) the following example of Seiberg–Witten Toda curves for $\mathcal{N}=2$ susy Yang–Mills with $G=\mathrm{SU}(N)$, $N=N_c$, no masses and moduli $u_k\in \mathcal{M}=$ quantum moduli space of inequivalent vacua:

$$\begin{array}{}\text{(a10)}& y^2=P^2-4{\mathrm{\Lambda }}^{2N},\end{array}$$

$$y={\mathrm{\Lambda }}^N(w-\frac{1}{w}),P={\lambda }^N-\sum _2^N{u}_k{\lambda }^{N-k}={\mathrm{\Lambda }}^N(w+\frac{1}{w}).$$

Here, $\lambda$ is the quantum scale, $\xi$ is a local coordinate at ${\mathrm{\infty }}_{\pm }$ with $\mathrm{\Lambda }\xi \sim w^{\mp (1/N)}$ with $w\to \mathrm{\infty }$ at ${\mathrm{\infty }}_{+}$ and $w\to 0$ at ${\mathrm{\infty }}_{-}$, and $g=N-1$. One defines

$$\begin{array}{}\text{(a11)}& d{\hat{\mathrm{\Omega }}}_n=P_{+}^{n/N}\left(\frac{dw}{w}\right)\end{array}$$

and

$$d{\mathrm{\Omega }}_n=d{\hat{\mathrm{\Omega }}}_n-\sum _{1}g\left({\oint }_{A_j}d{\hat{\mathrm{\Omega }}}_n\right)d{\omega }_j$$

($n<2N$ for technical reasons and $d{\omega }_j\sim$ holomorphic differentials). The standard Whitham theory is now based on

$$\begin{array}{}\text{(a12)}& dS=\sum _1^M{T}_{n}d{\hat{\mathrm{\Omega }}}_n=\sum _1^M{T}_{n}d{\mathrm{\Omega }}_n+\sum _1^g{\alpha }_{j}d{\omega }_j,\end{array}$$

where $M<2N$ and $T_0=0$ for $N_f=0$. One has then Whitham equations

$$\begin{array}{}\text{(a13)}& \frac{\partial d{\mathrm{\Omega }}_A}{\partial T_B}=\frac{\partial d{\mathrm{\Omega }}_B}{\partial T_A}\end{array}$$

with $\partial dS/\partial {\alpha }_j=d{\omega }_j$ and $\partial dS/\partial T_n=d{\omega }_n$ for $(T_n,{\alpha }_j)$ independent. The pre-potential $F$ arises via

$$\begin{array}{}\text{(a14)}& \frac{\partial F}{\partial {\alpha }_j}={\oint }_{B_j}dS\end{array}$$

and ${\partial }_{n}F=(1/2\pi in){\mathrm{Res}}_0{\xi }^{-n}dS$, where involves ${\mathrm{\infty }}_{\pm }$ and the Seiberg–Witten differential is

$$\begin{array}{}\text{(a15)}& d{S}_{SW}=d{\hat{\mathrm{\Omega }}}_1=\lambda \left(\frac{dw}{w}\right)=\lambda \frac{dP}{y}=\lambda \frac{dy}{P}.\end{array}$$

Thus, for $T_n={\delta }_{n,1}$ one has the Seiberg–Witten situation $F^{\text{SW}}=\tilde{F}$ and one writes then also $a_i={\alpha }_i$.

General framework.[edit]

The Whitham formulation of I. Krichever, developed in great detail with D.H. Phong (cf. [a43], [a44], [a45], [a46], [a47]), involves a Riemann surface ${\mathrm{\Sigma }}_g$ with $M$ punctures $P_{\alpha }$. One picks in an ad hoc manner two Abelian differentials $dE$ and $dQ$ having certain properties and sets $dS=QdE$ as a Seiberg–Witten-type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times $T_A$ with corresponding differentials $d{\mathrm{\Omega }}_A$. For suitable choices of $dE$ and $dQ$ the formulation is adequate for Seiberg–Witten-type situations and topological field theories with Witten–Dijkgraaf–Verlinde–Verlinde equations will arise as well.

Soft susy breaking.[edit]

There is another role for Whitham times, via (cf. [a26], [a55])

$$\begin{array}{}\text{(a16)}& {\hat{T}}_n=T_n{T}_1^{-1},{\hat{u}}_k=T_1^k{u}_k,\end{array}$$

and ${\hat{a}}_i={\alpha }_i(u_k,T_1,{\hat{T}}_{n>1}=0)=T_1{a}_i(u_k,\mathrm{\Lambda }=1)=a_i({\hat{u}}_k,\mathrm{\Lambda }=T_1)$ (note $T_1\sim \mathrm{\Lambda }$ in the Seiberg–Witten situation). Then one defines

$$\begin{array}{}\text{(a17)}& s_1=-i\log (\lambda )\end{array}$$

and $s_n=-i{\hat{T}}_n$ and these are promoted to spurion superfields ${\mathcal{S}}_n=s_n+{\theta }^2{F}_n$ and $V_n=(1/2)D_n{\theta }^2{\bar{\theta }}^2$ in $\mathcal{N}=1$ superfield language ($\theta$ and $\bar{\theta }$ are Grassmann variables while $D_n$ and $F_n$ are auxiliary fields). One has a family of non-susy theories and soft susy breaking $\mathcal{N}=2\to \mathcal{N}=0$ is achieved by fixing $s_n=0$ for $n>1$ and using $D_n$, $F_n$ ($n\ge 1$) as susy breaking parameters (actually, the $F_n$ alone will suffice). In any event, one can develop formulas involving $\lambda$, ${\tilde{T}}_n$ and ${\alpha }_j$ derivatives of the pre-potential and eventually parametrize soft susy breaking terms induced by all of the Casimirs.

Isomonodromy.[edit]

Various isomonodromy problems can be treated by multi-scale analysis to produce results indicating that isomonodromy deformations in WKB approximation correspond to modulation of isospectral problems (with Whitham-type equations as modulation equations). One can generate a pre-potential, period integrals, etc. as in Seiberg–Witten theory (see e.g. [a66], [a67], [a68], [a69], [a70]). There are also isomonodromy connections to the Knizhnik–Zamolodchikov–Bernard equations (cf. [a51], [a52], [a53], [a63]); these equations arise in various ways in conformal field theory, geometric quantization of flat bundles, etc. Here one takes $FB({\mathrm{\Sigma }}_g,G)$ as flat vector bundles over ${\mathrm{\Sigma }}_g$ with $G=\mathrm{GL}(N,\mathbf{C}\mathbf{)}$ and smooth connections $\mathcal{A}\sim (A,\bar{A})$. "Flat" means zero curvature and with an arbitrary $\kappa$ this has the form

$$\begin{array}{}\text{(a18)}& (\kappa \partial +A)\psi =0\end{array}$$

and $(\bar{\partial }+\bar{A})\psi =0$. Let $\mu \in {\mathrm{\Omega }}^{-1,1}({\mathrm{\Sigma }}_g)$ (Beltrami differentials), so $\mu =\mu (z,\overline{z}){\partial }_{\overline{z}}\otimes d\overline{z}$ and set , where $\text{l}=3g-3$ ($g>1$) and ${\mu }_a^0$ is a basis in $T{\mathcal{M}}_g$. Then (a18) becomes

$$\begin{array}{}\text{(a19)}& (\kappa \partial +A)\psi =0\end{array}$$

and $(\bar{\partial }+\mu \partial +\bar{A})\psi =0$. Let $\gamma$ be a homotopically non-trivial cycle in ${\mathrm{\Sigma }}_g$ such that $(z_0,{\bar{z}}_0)\in \gamma$ with $\psi (z_0,{\bar{z}}_0)=I$ and write $\mathcal{Y}(\gamma )=\psi (z_0,{\bar{z}}_0){|}_{\gamma }=P\exp ({\oint }_{\gamma }\mathcal{A})$ (path-ordered exponential), which yields a representation of ${\mathrm{\Pi }}_1({\mathrm{\Sigma }}_g,z_0)$ in $\mathrm{GL}(N,\mathbf{C})$. The independence of monodromy $\mathcal{Y}$ to complex structure deformation corresponds to for $a=1,\dots ,\text{l}$. Compatibility with (a19) requires

$$\begin{array}{}\text{(a20)}& {\partial }_{a}A=0\text{ and }\partial \bar{A}=(1/\kappa )A{\mu }_a^0.\end{array}$$

These equations are Hamiltonian when $FB({\sigma }_g,G)$ has a symplectic form ${\omega }^0=\int {\mathrm{\Sigma }}_g⟨\delta A,\delta \bar{A}⟩$ with Hamiltonians . Consider the bundle $\mathcal{P}$ over ${\mathcal{M}}_g$ with fibre $FB$ (using $(A,\bar{A},t\sim t_a)$ as local coordinates). A gauge fixing plus flatness corresponds to reduction from $FB\to \widetilde{FB}$ and one can (via WZW theory) fix the gauge to get a bundle $\tilde{\mathcal{P}}$ with fibre $\widetilde{FB}$ and equations

$$\begin{array}{}\text{(a21)}& (\kappa \partial +L)\psi =0\end{array}$$

with $(\bar{\partial }+\mu \partial +\bar{L})\psi =0$ and $(\kappa {\partial }_a+M_a)\psi =0$, where $M_a$ comes from the gauge transformation. Putting in the canonical form via local coordinates $(v_i,u_i)$ in $\widetilde{FB}$, where $i=1,\dots ,M=(N^2-1)(g-1)$, one can write

$$\begin{array}{}\text{(a22)}& {\omega }^0=(\delta v,\delta u)\end{array}$$

with $\omega ={\omega }^0-(1/\kappa )\sum \delta H_{\alpha }\delta t_{\alpha }$. Using the Poincaré–Cartan invariant form $\mathrm{\Theta }=(u,\delta v)-(1/\kappa )\sum H_a\delta t_a$ there exist $3g-3$ vector fields which annihilate $\mathrm{\Theta }$. With $\{.{\}}_0\sim {\omega }^0$-structure this gives

$$\begin{array}{}\text{(a23)}& \kappa {\partial }_s{H}_r-\kappa {\partial }_r{H}_s+\{H_s,H_r{\}}_0=0.\end{array}$$

These equations define flat connections in $\tilde{\mathcal{P}}$ and are referred to as a Whitham hierarchy of isomonodromic deformations. For a given $f(u,v,t)$ on $\tilde{\mathcal{P}}$ they take the form

$$\begin{array}{}\text{(a24)}& \frac{df}{d{t}_s}=\kappa {\partial }_{s}f+\{H_s,f\}\end{array}$$

and one can introduce a pre-potential $F$ on $\tilde{\mathcal{P}}$ giving Hamilton–Jacobi equations (cf. Hamilton–Jacobi theory)

$$\begin{array}{}\text{(a25)}& \kappa {\partial }_{s}F+H_s(\frac{\delta F}{\delta u},u,t)=0.\end{array}$$

Thus, one has a derivation of deformation equations, properly referred to as a Whitham hierarchy, which involves no averaging or multi-scale analysis. One can also compare the Baker–Akhiezer function $\psi$ in the Whitham hierarchy of isomonodromic deformations with elements of a certain Hitchin hierarchy (cf. also Hitchin system) using a WKB approximation with fast times $t_S^H$ and slow times $T_S\sim t_s$.

Contact terms.[edit]

For $\mathcal{N}=2$ susy gauge theory on a $4$-manifold with $b_{2+}=1$ there is a $u$-plane integral for, say, $\mathrm{SU}(N)$ situations, which can be related to a Toda theory with fast and slow (Whitham) times (cf. [a55], [a56], [a57], [a58], [a59], [a71], [a72]).

Witten–Dijkgraaf–Verlinde–Verlinde.[edit]

There is a beautiful and elaborate theory of B. Dubrovin and others based on Frobenius manifolds (cf. [a15], [a16], [a17], [a18], [a19], [a20], [a21], [a22], [a23], [a24]). This approach is especially pleasing since there is a great deal of motivation and natural structure. There are many connections to mathematics and physics and this approach has led to extensive development in Frobenius manifolds, quantum cohomology and $K$-theory, singularity theory, Witten–Dijkgraaf–Verlinde–Verlinde, etc. (see e.g. [a15], [a16], [a17], [a18], [a19], [a20], [a21], [a30], [a54]). A simple Hurwitz-space Korteweg–de Vries–Landau–Ginsburg model is as follows.

Let ${\mathcal{M}}_{g,n+1}$ be the moduli space of $g$ gap Korteweg–de Vries solutions based on $L={\partial }^{n+1}-q_1{\partial }^{n-1}-\dots -q_n$ with ramification based on $W=p^{n+1}-q_1{p}^{n-1}-\dots -q_n$. One defines Whitham times

$$\begin{array}{}\text{(a26)}& T_i=-\frac{n+1}{n+1-i}{\mathrm{Res}}_{\mathrm{\infty }}{W}^{1-[i/(n+1)]}dp,\end{array}$$

$$T_{n+\alpha }=\frac{1}{2\pi i}{\oint }_{A_{\alpha }}pdW,T_{g+n+\alpha }={\oint }_{B_{\alpha }}dp,$$

where $1\le i\le n$ and $1\le \alpha \le g$. These are flat times for a certain metric and determine a Whitham hierarchy, a Frobenius manifold and a topological field theory of Landau–Ginsburg type satisfying the Witten–Dijkgraaf–Verlinde–Verlinde equations (associativity equations for related field correlators).

References[edit]