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Article Contents

# On local well-posedness and ill-posedness results for a coupled system of mkdv type equations

• We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations

$\begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) = 0,&v(x,0) = \phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) = 0,& w(x,0) = \psi(x), \end{cases} \end{equation*}$

and prove the local well-posedness results for a given data in low regularity Sobolev spaces $H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R})$, $s,k> -\frac12$ and $|s-k|\leq 1/2$, for $\alpha\neq 0,1$. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $C^3$ at the origin, when $s<-1/2$ or $k<-1/2$ or $|s-k|>2$; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1/2$ (b) $k-2s>1$ or $s<-1/2$; (c) $s = k = -1/2$;

Mathematics Subject Classification: 35A01, 35M31, 35Q53.

 Citation:

• Figure 1.  Theorem 1.4

Figure 2.

Figure 3.

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