The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation: IV. Time dependent coefficients

In this paper, we consider an initial-value problem for the Korteweg-de Vries equation with time dependent coefficients. The normalized variable coefficient Korteweg-de Vries equation considered is given by \begin{equation*} u_{t}+ \Phi (t) u u_{x}+ \Psi (t) u_{xxx}=0, \quad -\infty <x<\infty , \quad t>0, \end{equation*} where $x$ and $t$ represent dimensionless distance and time respectively, whilst $\Phi (t)$, $\Psi (t)$ are given functions of $t (>0)$. In particular, we consider the case when the initial data has a discontinuous expansive step, where $u(x,0)=u_{+}$ for $x \ge 0$ and $u(x,0)=u_{-}$ for $x<0$. We focus attention on the case when $\Phi (t)=t^{\delta }$ (with $\delta >-\frac {2}{3}$) and $\Psi (t)=1$. The constant states $u_{+}$, $u_{-}$ ($<u_{+}$) and $\delta$ are problem parameters. The method of matched asymptotic coordinate expansions is used to obtain the large-$t$ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in $x \ge \frac {u_{-} }{(\delta +1)}t^{(\delta +1)}$ as $t \to \infty$, while the solution is oscillatory in $x<\frac {u_{-}}{(\delta +1)}t^{(\delta +1)}$ as $t \to \infty$. We conclude with a brief discussion of the structure of the large-$t$ solution of the initial-value problem when the initial data is step-like being continuous with algebraic decay as $|x| \to \infty$, with $u(x,t) \to u_{+}$ as $x \to \infty$ and $u(x,t) \to u_{-} (<u_{+})$ as $x \to -\infty$.