A note on the large-time development of the solution to an initial-value problem for the variable coefficient Korteweg-de Vries equation

In this article, we consider an initial-value problem for a variable coefficient Korteweg-de Vries equation.
The normalized variable coefficient Korteweg-de Vries equation considered is given by
u_t + uu_x + e^αt u_xxx = 0, −∞ < x < ∞, t > 0,
where x and t represent dimensionless distance and time respectively, and α(> 0) is a constant.
In particular, we consider the case when the initial data has a discontinuous expansive step, where
u(x, 0) = u+ for x ≥ 0 and u(x, 0) = u₋ for x < 0. The method of matched asymptotic coordinate
expansions is used to obtain the large-t asymptotic structure of the solution to this problem. We find that
the large-t attractor for the solution u(x, t) of the initial-value problem is based on the integral of the
standard Airy function, where
u(ze α/3 t , t) → [(u₋ + 2u₊)/3 + (u₊ − u₋)∫₀ ^(α/3)1/3z Ai(s) ds]
as t → ∞ with z = xe^{− α/3 t} = O(1). Further, this large-t attractor forms in a stretching frame of reference
of thickness x = O (e ^αt/3) as t → ∞.