<s>
In	O
mathematics	O
and	O
physics	O
,	O
specifically	O
the	O
study	O
of	O
field	O
theory	O
and	O
partial	O
differential	O
equations	O
,	O
a	O
Toda	B-Algorithm
field	I-Algorithm
theory	I-Algorithm
,	O
named	O
after	O
Morikazu	O
Toda	O
,	O
is	O
specified	O
by	O
a	O
choice	O
of	O
Kac	O
–	O
Moody	O
algebra	O
and	O
a	O
specific	O
Lagrangian	O
.	O
</s>
<s>
Toda	B-Algorithm
field	I-Algorithm
theories	I-Algorithm
are	O
classified	O
according	O
to	O
their	O
associated	O
Kac	O
–	O
Moody	O
algebra	O
.	O
</s>
<s>
Toda	B-Algorithm
field	I-Algorithm
theories	I-Algorithm
usually	O
refer	O
to	O
theories	O
with	O
a	O
finite	O
Kac	O
–	O
Moody	O
algebra	O
.	O
</s>
<s>
If	O
the	O
Kac	O
–	O
Moody	O
algebra	O
is	O
affine	O
,	O
it	O
is	O
called	O
an	O
affine	O
Toda	B-Algorithm
field	I-Algorithm
theory	I-Algorithm
(	O
after	O
the	O
component	O
of	O
which	O
decouples	O
is	O
removed	O
)	O
.	O
</s>
<s>
If	O
it	O
is	O
hyperbolic	O
,	O
it	O
is	O
called	O
a	O
hyperbolic	O
Toda	B-Algorithm
field	I-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
Toda	B-Algorithm
field	I-Algorithm
theories	I-Algorithm
are	O
integrable	O
models	O
and	O
their	O
solutions	O
describe	O
solitons	O
.	O
</s>
