The Symmetric Regularized-Long-Wave Equation: Ill-posedness and Long Period Limit

In the present work we obtain two important results for the Symmetric Regulraized-Long-Wave equation. First we prove that the initial value problem for this equation is ill-posed for data in $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}),$ if $s< 0,$ in the sense that the flow-map cannot be continuous at the origin from $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$ to even $(\mathcal{D}'(\mathbb{R}))^2.$ We also establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.