## Abstract

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

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

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

as t → ∞ with z = xe

of thickness x = O (e

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 coordinateexpansions 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 referenceof thickness x = O (e

^{αt/3}) as t → ∞.Original language | English |
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Pages (from-to) | 717-725 |

Journal | IMA Journal of Applied Mathematics |

Volume | 82 |

Issue number | 4 |

Early online date | 18 May 2017 |

DOIs | |

Publication status | Published - 1 Aug 2017 |

## Keywords

- Korteweg-de Vries equation
- asymptotic methods
- non-linear waves