- #1
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The KP hierarchy is obtained the following way:
First we introduce the pseudo-differential operator
## L = \partial + \sum_{i=1}^\infty a_i(t_1,t_2 \ldots) \partial^{-i} ##
and then imposing the Lax equations
## \frac{\partial L}{\partial t_i} = [(L^i)_+,L] ##
for each i=1,2, etc (and where ##t_1\equiv x##).
(My source is Glimpses of Soliton theory, p.227).
My problem is that the Lax equation for i=1 does not make sense to me. If we set i=1, the left side contains the second order operator ##\partial^2## whereas in the commutator the ##\partial^2## cancel out, so there is no such term on the right. Maybe the authors made a mistake by saying that we must include i=1? I know that in order to recover the KP equation we use the equations for i=2 and i=3 but not i=1, so maybe thee is a typo and they really meant to consider i=2,3... ?
First we introduce the pseudo-differential operator
## L = \partial + \sum_{i=1}^\infty a_i(t_1,t_2 \ldots) \partial^{-i} ##
and then imposing the Lax equations
## \frac{\partial L}{\partial t_i} = [(L^i)_+,L] ##
for each i=1,2, etc (and where ##t_1\equiv x##).
(My source is Glimpses of Soliton theory, p.227).
My problem is that the Lax equation for i=1 does not make sense to me. If we set i=1, the left side contains the second order operator ##\partial^2## whereas in the commutator the ##\partial^2## cancel out, so there is no such term on the right. Maybe the authors made a mistake by saying that we must include i=1? I know that in order to recover the KP equation we use the equations for i=2 and i=3 but not i=1, so maybe thee is a typo and they really meant to consider i=2,3... ?