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 u t + Φ ( t ) u u x + Ψ ( t ) u x x x = 0 , − ∞ > x > ∞ , t > 0 , \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 x and t t represent dimensionless distance and time respectively, whilst Φ ( t ) \Phi (t) , Ψ ( t ) \Psi (t) are given functions of t ( > 0 ) t (>0) . In particular, we consider the case when the initial data has a discontinuous expansive step, where u ( x , 0 ) = u + u(x,0)=u_{+} for x ≥ 0 x \ge 0 and u ( x , 0 ) = u − u(x,0)=u_{-} for x > 0 x>0 . We focus attention on the case when Φ ( t ) = t δ \Phi (t)=t^{\delta } (with δ > − 2 3 \delta >-\frac {2}{3} ) and Ψ ( t ) = 1 \Psi (t)=1 . The constant states u + u_{+} , u − u_{-} ( > u + >u_{+} ) and δ \delta are problem parameters. The method of matched asymptotic coordinate expansions is used to obtain the large- t t asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ≥ u − ( δ + 1 ) t ( δ + 1 ) x \ge \frac {u_{-} }{(\delta +1)}t^{(\delta +1)} as t → ∞ t \to \infty , while the solution is oscillatory in x > u − ( δ + 1 ) t ( δ + 1 ) x>\frac {u_{-}}{(\delta +1)}t^{(\delta +1)} as t → ∞ t \to \infty . We conclude with a brief discussion of the structure of the large- t t solution of the initial-value problem when the initial data is step-like being continuous with algebraic decay as | x | → ∞ |x| \to \infty , with u ( x , t ) → u + u(x,t) \to u_{+} as x → ∞ x \to \infty and u ( x , t ) → u − ( > u + ) u(x,t) \to u_{-} (>u_{+}) as x → − ∞ x \to -\infty .