EXISTENCE OF TRAVELLING WAVES IN THE FRACTIONAL BURGERS EQUATION

Abstract

We construct travelling waves in the Burgers equation with the fractional Laplacian $(D^{2})^{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D6FC}\in (1/2,1)$. This is done by first constructing odd solutions $u_{\unicode[STIX]{x1D700}}$ of $uu^{\prime }=K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u+\unicode[STIX]{x1D700}_{2}u^{\prime \prime }$, $u(-\infty )=u_{c}>0$, with $K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u$ nonsingular, and then passing to the limit $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2}\rightarrow 0$, to give $K_{\unicode[STIX]{x1D700}_{1}}\ast u_{\unicode[STIX]{x1D700}}-k_{\unicode[STIX]{x1D700}_{1}}u_{\unicode[STIX]{x1D700}}\rightarrow (D^{2})^{\unicode[STIX]{x1D6FC}}u_{0}$ pointwise. The proof relies on operator splitting.