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Consider the BBM equation: $-u_{txx}+u_{t}=u_{x},\quad u(x,0)=u_0(x),\quad x,t\in{\bf R}$

One may rewrite it as

$u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ exists.

Here are my questions:

  1. Does $(I-A)^{-1}$ always exist?
  2. Is there an integral operator $K:L^2({\bf R})\to L^2({\bf R})$ such that $K=((I-A)^{-1}\partial_x)$?

1 Answers 1

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On $L^2(\bf R)$, yes, $(I - \frac{d^2}{dx^2})^{-1}$ exists. It corresponds via the Fourier transform to multiplication by $1/(1 + p^2)$, and your $K$ corresponds to multiplication by $i p/(1 + p^2)$, or convolution with the inverse Fourier transform of that, namely $i {\rm sgn}(x) e^{-|x|}/2$.