### Abstract

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

Original language | English |
---|---|

Article number | 036605 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 86 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Sep 18 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*86*(3), [036605]. https://doi.org/10.1103/PhysRevE.86.036605

}

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 86, no. 3, 036605. https://doi.org/10.1103/PhysRevE.86.036605

**Undular bore theory for the Gardner equation.** / Kamchatnov, A. M.; Kuo, Y. H.; Lin, T. C.; Horng, T. L.; Gou, S. C.; Clift, R.; El, G. A.; Grimshaw, R. H.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Undular bore theory for the Gardner equation

AU - Kamchatnov, A. M.

AU - Kuo, Y. H.

AU - Lin, T. C.

AU - Horng, T. L.

AU - Gou, S. C.

AU - Clift, R.

AU - El, G. A.

AU - Grimshaw, R. H.J.

PY - 2012/9/18

Y1 - 2012/9/18

N2 - We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

AB - We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

UR - http://www.scopus.com/inward/record.url?scp=84866910955&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevE.86.036605

DO - 10.1103/PhysRevE.86.036605

M3 - Article

AN - SCOPUS:84866910955

VL - 86

JO - Physical review. E

JF - Physical review. E

SN - 1539-3755

IS - 3

M1 - 036605

ER -