Moutard transformation

A mapping of the same type as the Darboux transformation: it connects the solutions and the coefficients of equations

\begin{equation*} \psi _ { x y } + u ( x , y ) \psi = 0 \end{equation*}

so that if $\varphi$ and $\psi$ are different solutions of it, then the solution of the twin equation with $\psi \rightarrow \psi [ 1 ]$, $u ( x , y ) \rightarrow u [ 1 ] ( x , y )$ may be constructed as the solution of the system

\begin{equation*} ( \psi [ 1 ] \varphi ) _ { x } = - \varphi ^ { 2 } ( \psi \varphi ^ { - 1 } ) _ { x }, \end{equation*}

\begin{equation*} ( \psi [ 1 ] \varphi ) _ y = \varphi ^ { 2 } ( \psi \varphi ^ { - 1 } ) _ y. \end{equation*}

The transformed coefficient (a potential in mathematical physics) is given by

\begin{equation*} u [ 1 ] = u - 2 ( \operatorname { log } \varphi ) _ { x y } = - u + \frac { \varphi _ { x } \varphi_y } { \varphi ^ { 2 } }; \end{equation*}

in other words,

\begin{equation*} \psi [ 1 ] = \psi - \frac { \varphi \Omega ( \varphi , \psi ) } { \Omega ( \varphi , \varphi ) }, \end{equation*}

where $\Omega$ is the integral of the exact differential form

\begin{equation*} d \Omega = \varphi \psi _ { x } d x + \psi \varphi_y d y. \end{equation*}

The important feature defining it is that the transform is parametrized by a pair of solutions of the equation and that the transform vanishes if the solutions coincide.

Clearly, the Moutard equation can be transformed to a $2$-dimensional Schrödinger equation (cf. also Schrödinger equation), and can be studied in connection with the central problems of classical differential geometry. In the theory of solitons (cf. Soliton) it enters via Lax pairs for non-linear equations as the Nizhik–Veselov–Novikov equations [a1], [a2].

In [a3] the Moutard transformation appears in the context of Painlevé analysis.

There is a generalization of Moutard transformations to higher dimensions [a4]; a proof of the (local) completeness can be found in [a5].

References

[a1]	C. Athorne, J.J.C. Nimmo, "On the Moutard transformation for integrable partial differential equations" Inverse Problems , 7 (1991) pp. 809–826
[a2]	V.B. Matveev, M.A. Salle, "Darboux transformations and solitons" , Springer (1991)
[a3]	P.G. Estevez, S. Leble, "A wave equation in $2 n + 1$: Painlevé analysis and solutions" Inverse Problems , 11 (1995) pp. 925–937
[a4]	C. Athorne, "On the characterization of Moutard transformations" Inverse Problems , 9 (1993) pp. 217–232
[a5]	E. Gahzha, "On completeness of the Moutard transformations" solv-int@xyz.lanl.gov , 9606001 (1996)