The sine-Gordon equation Bäcklund transform

suppose u solution of sine-gordon equation

u

x
y


=
sin
⁡
u
.



{\displaystyle u_{xy}=\sin u.\,}

then system

v

x





=

u

x


+
2
a
sin
⁡


(





u
+
v

2




)







v

y





=
−

u

y


+


2
a


sin
⁡


(





v
−
u

2




)










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

where arbitrary parameter, solvable function v satisfy sine-gordon equation. example of auto-bäcklund transform.

by using matrix system, possible find linear bäcklund transform solutions of sine-gordon equation.