# The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

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## Abstract

**:**

## 1. Introduction

## 2. Analysis

#### 2.1. The Propagation of Wave Packets in Time and Space

**same**canonical form. Thus, in case A, we set

#### 2.2. The Evolution of Specific NLS Solutions

## 3. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

NLS | Nonlinear Schrödinger equation |

MI | Modulation instability |

AB | Akhmediev breather |

## References

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**Figure 1.**(

**a**) spatiotemporal evolution of the modulus of the soliton $|{A}_{S}(x,t)|$ for the carrier parameters $ak=0.1$ and $a=0.01$ m; (

**b**) spatiotemporal evolution of the modulus of the soliton $|{B}_{S}(x,t)|$ for the carrier parameters $ak=0.1$ and $a=0.01$ m.

**Figure 2.**(

**a**) spatiotemporal evolution of the modulus of the Peregrine breather $|{A}_{P}(x,t)|$ for the carrier parameters $ak=0.1$ and $a=0.01$ m; (

**b**) spatiotemporal evolution of the modulus of the Peregrine breather $|{B}_{P}(x,t)|$ for the carrier parameters $ak=0.1$ and $a=0.01$ m.

**Figure 3.**(

**a**) spatiotemporal evolution of the modulus of an Kuznetsov–Ma breather $\left|{A}_{KM}\left(x,t\right)\right|$ for $\phi =0.88$ and the carrier parameters $ak=0.1$ and $a=0.01$ m; (

**b**) spatiotemporal evolution of the modulus of an Kuznetsov breather $\left|{B}_{KM}\left(x,t\right)\right|$ for $\phi =0.88$ and the carrier parameters $ak=0.1$ and $a=0.01$ m.

**Figure 4.**(

**a**) spatiotemporal evolution of the modulus of an AB $\left|{A}_{A}\left(x,t\right)\right|$ for $\phi =0.46$ and the carrier parameters $ak=0.1$ and $a=0.01$ m; (

**b**) spatiotemporal evolution of the modulus of an AB $\left|{B}_{A}\left(x,t\right)\right|$ for $\phi =0.46$ and the carrier parameters $ak=0.1$ and $a=0.01$ m.

**Figure 5.**(

**a**) temporal surface variation of an AB ${A}_{A}(0,t)$ for $\phi =0.46$ and background parameters $\alpha =0.1$ and $a=0.01$ m; (

**b**) temporal water surface variation of an AB ${B}_{A}(0,t)$ for $\phi =0.46$ and background parameters $\alpha =0.1$ and $a=0.01$ m.

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Chabchoub, A.; Grimshaw, R.H.J.
The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time. *Fluids* **2016**, *1*, 23.
https://doi.org/10.3390/fluids1030023

**AMA Style**

Chabchoub A, Grimshaw RHJ.
The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time. *Fluids*. 2016; 1(3):23.
https://doi.org/10.3390/fluids1030023

**Chicago/Turabian Style**

Chabchoub, Amin, and Roger H. J. Grimshaw.
2016. "The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time" *Fluids* 1, no. 3: 23.
https://doi.org/10.3390/fluids1030023