# Kdv solution solitons Bilinear Operator

## Homework Statement

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I'm solving for ##f_1## from ##B(f_{1}.1+1.f_{1})## from ## \frac{\partial}{\partial x}(\frac{\partial}{\partial t}+\frac{\partial^{3}}{\partial x^{3}})f_n=-\frac{1}{2}\sum^{n-1}_{m=1}B(f_{n-m}.f_{m}) ##

where ##B=D_tD_x+D_x^4##, where ##B## is the Bilinear operator.

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(above)

## The Attempt at a Solution

I get ##B(f_{1}.1+1.f_{1})=2f_{1xt}## not ##0!##. Whereas the method gets ##\frac{\partial}{\partial x}(\frac{\partial}{\partial t}+\frac{\partial^{3}}{\partial x^{3}})f_1=0##

Well a general solution to the equation of this form is $$f_1 = f(x^3-t)$$ but I am still not sure what you are looking for here.