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Asymptotics of solutions of a modified Whitham equation with surface tension
P. I. Naumkin National Autonomous University of Mexico, Institute of Mathematics
Abstract:
We study the large-time behaviour of solutions of the Cauchy problem
for 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 given
by the symbol
$$
\Lambda (\xi)=a^{-{1}/{2}}\xi
\biggl(\sqrt{(1+a^2\xi^2) \frac{\operatorname{tanh}a\xi}{a\xi}\,}-1\biggr)
$$
with a parameter
$a>0$. This symbol corresponds to the total dispersion
relation 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 $\mathbf{H}^{\nu}(\mathbb{R}) \cap
\mathbf{H}^{0,1}(\mathbb{R})$,
$\nu \geqslant 22$,
we prove the existence of a global-in-time solution and describe its
large-time asymptotic behaviour.
Keywords:
Whitham equation, critical non-linearity, large-time asymptotics.
UDC:
517.956.8 +
517.953
MSC: 35B40,
35Q35,
76B15 Received: 20.03.2017
Revised: 27.08.2018
DOI:
10.4213/im8673
Fulltext:
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