Asymptotics of solutions of a modified Whitham equation with surface tension

We study the large-time behaviour of solutions of the Cauchy problemfor a modified Whitham equation,$$\begin{cases}u_{t}+i\mathbf{\Lambda}u-\partial_{x}u^3=0, &(t,x) \in\mathbb{R}^2,u(0,x)=u_0(x), &x\in \mathbb{R},\end{cases}$$where the pseudodifferential operator $\mathbf{\Lambda}\equiv \Lambda(-i\partial_{x})=\mathcal{F}^{-1}[\Lambda (\xi) \mathcal{F}]$ is givenby the symbol$$\Lambda (\xi)=a^{-{1}/{2}}\xi(\sqrt{(1+a^2\xi^2) \frac{\operatorname{tanh}a\xi}{a\xi} }-1)$$with a parameter $a>0$. This symbol corresponds to the total dispersionrelation for water waves taking surface tension into account. Assuming that the total mass of the initial data is equal to zero($\int_{\mathbb{R}}u_0(x) dx=0$) and the initial data $u_0$ are small in the norm of $\mathbf{H}^{\nu}(\mathbb{R}) \cap\mathbf{H}^{0,1}(\mathbb{R})$, $\nu \geq 22$,we prove the existence of a global-in-time solution and describe itslarge-time asymptotic behaviour.

Whitham equation, critical non-linearity, large-time asymptotics