A first-order Fourier integrator for the nonlinear Schrödinger equation on 𝕋 without loss of regularity

2021 ◽  
Author(s):  
Yifei Wu ◽  
Fangyan Yao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Loss Of Regularity

Darboux transformation of the second-type nonlocal derivative nonlinear Schrödinger equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950123 ◽  
Author(s):  
De-Xin Meng ◽  
Kuang-Zhong Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Darboux Transformations ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation

The second-type nonlocal derivative nonlinear Schrödinger (NDNLSII) equation is studied in this paper. By constructing its [Formula: see text]-order Darboux transformations (DT) from the first-order DT, Vandermonde-type determinant solutions of the NDNLSII equation are obtained from zero seed solutions, which would be singular unless the square of eigenvalues are purely imaginary.

Higher-Order Rogue Waves for a Fifth-Order Dispersive Nonlinear Schrödinger Equation in an Optical Fibre

2015 ◽  
Vol 70 (5) ◽  
pp. 365-374 ◽  
Author(s):  
Qi-Min Wang ◽  
Yi-Tian Gao ◽  
Chuan-Qi Su ◽  
Yu-Jia Shen ◽  
Yu-Jie Feng ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Fifth Order

AbstractIn this article, a fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an optical fibre. Rogue wave solutions are derived by virtue of the generalised Darboux transformation. Rogue wave structures and interaction are discussed through (i) the analyses on the higher-order rogue waves, the cubic, quartic, quintic, group-velocity, and phase-parameter effects; (ii) a higher-order rogue wave consisting of the first-order rogue waves via the interaction; (iii) characteristics of the rogue waves which are summarised, including the maximum/minimum values of the rogue waves and the number of the first-order rogue waves for composing the higher-order rogue wave; and (iv) spatial-temporal patterns which are illustrated and compared with those of the ‘self-focusing’ nonlinear Schrödinger equation. We find that the quintic terms increase the time of appearance for the first-order rogue waves which form the higher-order rogue wave, and that the quintic terms affect the interaction among the first-order rogue waves, which elongates the distance of appearance for the higher-order rogue wave.

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