### Abstract

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite-order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found.

Language | English |
---|---|

Pages | 359-384 |

Number of pages | 26 |

Journal | Studies in Applied Mathematics |

Volume | 142 |

Issue number | 3 |

Early online date | 10 Jan 2019 |

DOIs | |

Publication status | Published - Apr 2019 |

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### Keywords

- asymptotic analysis
- solitons and integrable systems

### Cite this

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*Studies in Applied Mathematics*, vol. 142, no. 3, pp. 359-384. https://doi.org/10.1111/sapm.12252

**Generalized solitary waves in a finite-difference Korteweg-de Vries equation.** / Joshi, N.; Lustri, C. J.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Generalized solitary waves in a finite-difference Korteweg-de Vries equation

AU - Joshi, N.

AU - Lustri, C. J.

PY - 2019/4

Y1 - 2019/4

N2 - Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite-order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found.

AB - Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite-order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found.

KW - asymptotic analysis

KW - solitons and integrable systems

UR - http://www.scopus.com/inward/record.url?scp=85059850973&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/FL120100094

U2 - 10.1111/sapm.12252

DO - 10.1111/sapm.12252

M3 - Article

VL - 142

SP - 359

EP - 384

JO - Studies in Applied Mathematics

T2 - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 3

ER -