Self-similar asymptotics for solutions to the intermediate long-wave equation

Abstract

We consider the Cauchy problem for the intermediate long-wave equation $$\begin{aligned} u_{t}-\partial _{x}u^{2}+\frac{1}{\vartheta }u_{x}+VP\int _{\mathbb {R}}\frac{1}{2\vartheta }\coth \left( \frac{\pi \left( y-x\right) }{2\vartheta }\right) u_{yy}\left( t,y\right) \mathrm{d}y=0, \end{aligned}$$ where $$\vartheta >0$$ . Our purpose in this paper is to prove the large time asymptotic behavior of solutions under the nonzero mass condition $$\int u_{0}\left( x\right) \mathrm{d}x\ne 0$$ .