### Abstract

The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

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

Article number | 042302 |

Journal | Physics of Plasmas |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 11 2009 |

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### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics

### Cite this

*Physics of Plasmas*,

*16*(4), [042302]. https://doi.org/10.1063/1.3093383

**The truncation model of the derivative nonlinear Schrödinger equation.** / Sánchez-Arriaga, G.; Hada, Tohru; Nariyuki, Y.

Research output: Contribution to journal › Article

*Physics of Plasmas*, vol. 16, no. 4, 042302. https://doi.org/10.1063/1.3093383

}

TY - JOUR

T1 - The truncation model of the derivative nonlinear Schrödinger equation

AU - Sánchez-Arriaga, G.

AU - Hada, Tohru

AU - Nariyuki, Y.

PY - 2009/5/11

Y1 - 2009/5/11

N2 - The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

AB - The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

UR - http://www.scopus.com/inward/record.url?scp=65449146365&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65449146365&partnerID=8YFLogxK

U2 - 10.1063/1.3093383

DO - 10.1063/1.3093383

M3 - Article

AN - SCOPUS:65449146365

VL - 16

JO - Physics of Plasmas

JF - Physics of Plasmas

SN - 1070-664X

IS - 4

M1 - 042302

ER -