THE KORTEWEG-DE VRIES EQUATION

The Korteweg-de Vries equation

$$u_t + u_{xxx} + 3uu_x = 0$$ (16)

has meromorphic singularities of the form

$$u = \phi^{-2}\sum_{j=0}^\infty u_j \phi^j$$ (17)

about the singularity manifold $\phi(x,t) = 0$. In the above we find that $\phi$ , $u_4$ and $u_6$ are arbitrary functions, and it is required that, for (17) to be well defined, $\phi$ be non-characteristic. That is, $\phi_x \ne 0$ when $\phi = 0$. For locally analytic data we will show that (17) converges in a neighborhood of $\phi = 0$.

It is also of interest to consider the slight generalization of (16)

$$(\partial/\partial x) (u_t + u_{xxx} + 3uu_x ) = 0$$ (18)

which has the expansion (17) with arbitrary functions $\phi$, $u_4$, $u_5$ and $u_6$.

Now, we truncate (17) after the constant term to obtain

$$u = u_0/\phi^2 + u_1/\phi + u_2.$$ (19)

Substitution into (16), or (18), obtains a system of four equations in the four functions $\phi, u_0, u_1, u_2$. This is most readily seen for (18) since setting the arbitrary functions $u_4 = u_5 = u_6 = 0$ and requiring $u_3 = 0$ obtains (19) and four equations. From these we find

$$u = -4\phi_x^2/\phi^2 + 4\phi_{xx}/\phi + u_2$$ (20)

$$u_2 + \lambda = -(\phi_{xxx}/\phi_x) + \frac12(\phi_{xx}/\phi_x)^2$$ (21)

$$\phi_t/\phi_x + \{\phi;x\} = \lambda$$ (22)

where $u$ and $u_2$ satisfy the KdV equation. Again,

$$\{\phi;x\} = \phi_{xxx}/\phi_x - \frac32(\phi_{xx}/\phi_x)^2$$ (23)

, the Schwarzian derivative [7,8], is the unique differential invariant [19] of the Moebius group

$$\phi = (a\psi + b)/(c\psi + d).$$ (24)

The Moebius group is the unique group of conformal (monodromny preserving) automorphisms of the complex $\phi$ (Riemann) sphere.

The relations (21) and (22) imply that $u_2$ satisfy (16) since

$$u_{2,t} = -(\frac{\partial}{\partial x} + V)\frac{\partial}{\... ...ial x} (\frac{\partial}{\partial x} - V) \frac{\phi_t}{\phi_x}$$

where

$$V = \frac{\phi_{xx}}{\phi_x}.$$ (25)

This definition of $V$ obtains the modified KdV equation

$$V_t + V_{xxx} - \frac32 V^2V_x= \lambda V$$ (26)

from (21).

The Bäcklund-Darboux equation (20) may be written in the form

$$u = 4\frac{\partial^2}{\partial x^2} \ln \phi + u_2$$ (27)

where

$$u_2 + \lambda = -\frac{\partial}{\partial x} (\phi_{xx}/\phi_x) - \frac12(\phi_{xx}/\phi_x)^2$$ (28)

and $\phi$ satisfies the Schwarzian-KdV equation (22).

We regard (28) as a Miura transformation from (22) to (16). It also has the form of a Ricati equation in the variable $W = \phi_{xx}/\phi_x$ and can be linearized by the substitution $W = -2v_x/v$. This obtains the linear equation for $v$

$$2v_{xx} = (u_2 + \lambda)v$$ (29)

and the identification $\phi_x = v^{-2}$. From (22) the additional linear equation

$$v_t = (2\lambda - u_2)v_x + \frac12u_{2,x}v$$ (30)

is found. By construction (29) and (30) are the Lax pair for the KdV equation and imply $u_2$ is a KdV solution.

The linearizing substitution for $\phi$ has the form

$$\phi = v_1/v_2$$

where $v_1$ and $v_2$ are solutions of (29).

In terms of the linear equation (29) the Bäcklund transformation is the classical Darboux transformation for adding elements to the spectra [20,21].

Consider next the Bäcklund-Darboux transformation for the modified KdV equation (26)

$$V=2\frac{\phi_x}{\phi} + V_1$$

where

$$V_1=-\frac{\phi_{xx}}{\phi_x},$$

and $\phi$ is a solution of (22). Comparing (25), (26) to the above obtains the discrete symmetry of the modified equation

$$V \Rightarrow -V.$$

Now, consider the Bäcklund transformations for the Schwarzian KdV equation (22). From the invariance of the Schwarzian derivative we have the invariance of (22) under the Moebius group (24). This invariance is also found by examining the singularities of (22), (8), and finding that the truncated expansion

$$\phi = \psi^{-1}\phi_0 + \phi_1$$ (31)

requires constant $\phi_0, \phi_1$ and $\psi$ satisfies (22). In other words, we find the Moebius invariance.

An additional Bäcklund transformation is found from the discrete symmetries of the modified equation (26). That is, $V \Rightarrow -V$ implies the invariant transformation

$$\phi_x = 1/\psi_x$$ (32)

for (22). The time dependent form of this transformation is found from (22) and (32). That is,

$$\phi_t/\phi_x + \psi_t/\psi_x + (\phi_{xx}/\phi_x)(\psi_{xx}/\psi_x) = 2\lambda$$ (33)

and (32) imply by $\phi_{xt}=\phi_{tx}$ and $\psi_{xt}=\psi_{tx}$ that $\phi$ and $\psi$ both satisfy (22). Therefore, we have a strong Bäcklund transformation for (22).

In section 4 we will show how the periodic fixed points of the Bäcklund transformations for (22) define finite dimensional invariant manifolds and as commuting hamiltonian flows factor the KdV flow on this invariant manifold.

Using the Bäcklund transformations, it is simple to show that the Laurent expansion

$$\phi = \xi^{-1}\phi_0 + \phi_1 + \cdots$$

converges in a neighborhood of $\xi(x,t) = 0$. The symmetry $\psi = \phi^{-1}$ maps the pole into a simple zero and the data satisfies the conditions of the Cauchy-Kovalevsky Theorem, being non-characteristic and locally analytic. Therefore, both expansions converge in a (punctured) neighborhood. Furthermore, the symmetry (32) maps the characteristic, non-conformal singularity

$$\phi = f(t) + \xi^3\phi_3 + \cdots$$

into a simple pole

$$\psi = \xi^{-1}\psi_0 + \cdots$$

and by the previous argument , this also converges. Using this result and the relation between the modified and KdV systems, the pole singularities for the KdV equation also converge. The characteristic singularities for the KdV are trivial.

The convergence of the Laurent series for the KdV equation

$$u=\phi^{-2}\sum_{j=0}^{\infty}u_j\phi^j$$

implies, by the recursion relations, the existence of the conservation laws

$$\frac{\partial}{\partial t} A_j = \frac{\partial}{\partial x}B_j$$

where

$$A_j=\sum_{k=0}^{\infty}\frac{(j+k)!}{j!k!}u_{j+2+k}\phi^k.$$

Therefore the Painlevé property directly implies the existence of formal integrals.

The KdV (16) and modified KdV (26) have a hamiltonian structure [9,10]

$$u_t + \frac{\partial}{\partial x} (u_{xx} + \frac32u^2) = 0$$ (34)

$$V_t + \frac{\partial}{\partial x} (V_{xx} -\frac12V^3) = 0$$ (35)

and are connected by the Miura transformation

$$u = \pm V_x - \frac12 V^2.$$ (36)

Note

$$u_{xx} + \frac32u^2 = \delta_u\int (-u_x^2 + \frac12 u^3) dx$$

$$V_{xx} - \frac12V^3 = \delta_V\int (-V_x^2 - \frac18 V^4) dx .$$

Using the Miura transformation finds the second hamiltonian structure for the KdV equation from the first hamiltonian structure of (35). By the change of variable formula $\Omega= (\delta_Vu) \partial_x (\delta_Vu)^t$. It is

$$\Omega = \partial^3_x + 2u\partial_x + u_x = (\partial_x - V)(\partial_x)(\partial_x + V)$$ (37)

where $u = V_x -\frac12V^2$ and the KdV equation is

$$u_t + \Omega \delta_u H_2 = 0$$

where $H_1 = \int \frac12 u^2$.

The gradient of the integrals for the KdV equation satisfy the Lenard recursion formula

$$\partial_x \delta_u H_{j+1} = \Omega \delta_u H_j$$ (38)

and the higher order equations are

$$u_t + \partial_x \delta_u H_j = 0$$ (39)

for $j=1,2,\cdots$. Now, putting together the above we have the following result [9].

The sequence of higher-order KdV equations

$$u_t + \partial_x \delta_u H_{j+1} = 0$$ (40)

for $j=1,2,3,\cdots$ has the Bäcklund-Darboux transformation

$$u = 4\frac{\partial^2}{\partial x^2} \ln \phi + u_2$$ (41)

where

$$u_2 = -\frac{\partial}{\partial x}(\phi_{xx}/\phi_x) - \frac12(\phi_{xx}/\phi_x)^2$$ (42)

and

$$\frac{\phi_t}{\phi_x} + \delta_u H_j(\{\phi;x\})=0.$$ (43)

Furthermore,

$$u_3 = \{\phi;x\}$$

and $u_2$ satisfy (40) and the sequence (43) is invariant under the Moebius group and the symmetry (32).

Note that the first few gradients are:

$$\delta_u H_1 = u$$

$$\delta_u H_2 = u_{xx} + \frac32u^2$$

$$\delta_u H_3 = u_{xxxx} + 5uu_{xx} + \frac52u_x^2 + \frac52u^3.$$

Now, using a simple leading order argument and the Bäcklund transformations to raise and lower the weight of the Laurent expansions it is not difficult to see that the higher-order systems must have the Painlevé Property and the formal Laurent expansions converge in a punctured neighborhood of the singularity manifold when the data is locally analytic in this neighborhood [9]. That is, the highest weight singularity for the $N^{th}$ Schwarzian equation is of the form

$$\phi = f(t) + \xi^{2N+1}\phi_{2N+1} + \cdots.$$

The odd order poles are, for $k < N$,

$$\phi=\xi^{-2k-1}\phi_0 + \cdots$$

and the odd order zeroes are, for $k < N$,

$$\phi=\xi^{2k+1}\phi_0 + \cdots.$$

The symmetry $\phi=1/\psi$ maps poles $\Leftrightarrow$ zeroes. The symmetry $\phi_x = 1/\psi_x$ maps a zero of order $2k+1$ $\Leftrightarrow$ a pole of order $2k-1$. Combining the two transformations any singularity can be mapped into a simple zero $\phi = \xi\phi_0 + \cdots$ and by the Cauchy-Kovalevsky Theorem this is single-valued and convergent. The map from this form into the complete set of singularities can be shown to introduce no multiple-valued $\ln\xi$ terms and to depend on the maximum number of allowed arbitrary functions.

Therefore, the KdV sequence has the Painlevé Property with convergent Laurent series.

To summarize, the Painlevé expansion truncated after the constant level term defines a form of modified equation that is expressed in terms of the Schwarzian derivative. By linearizing the Miura transformation the Lax pair is found. The discrete symmetries of the modified equations and the Moebius group are Bäcklund transformations for the Schwarzian equations. The Miura transformation allows the calculation of second hamiltonian structures and the associated recursion operators for the gradients of conserved densities. The action of Bácklund transformations on the singularity structure of equation sequences allows the conclusion that the sequence is Painlevé and the formal Laurent expansions converge.