The Liouville equation Bäcklund transform

a bäcklund transform can turn non-linear partial differential equation simpler, linear, partial differential equation.

for example, if u , v related via bäcklund transform

v

x





=

u

x


+
2
a
exp
⁡


(





u
+
v

2




)







v

y





=
−

u

y


−


1
a


exp
⁡


(





u
−
v

2




)










{\displaystyle {\begin{aligned}v_{x}&=u_{x}+2a\exp {\bigl (}{\frac {u+v}{2}}{\bigr )}\\v_{y}&=-u_{y}-{\frac {1}{a}}\exp {\bigl (}{\frac {u-v}{2}}{\bigr )}\end{aligned}}\,\!}

where arbitrary parameter, , if u solution of liouville equation

u

x
y


=
exp
⁡
u




{\displaystyle u_{xy}=\exp u\,\!}

then v solution of simpler equation,




v

x
y


=
0


{\displaystyle v_{xy}=0}

, , vice versa.

we can solve (non-linear) liouville equation working simpler linear equation.