Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions.

By Boling Guo, Liming Ling, Q. P. Liu. Physical review. E, Statistical, nonlinear, and soft matter physics, 2011

880 citations recorded by the source corpus.

In this paper, we construct a generalized Darboux transformation for the nonlinear Schrödinger equation. The associated N-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the Nth-order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7b3284bb94815b76e845432f6486078cb798c781

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Modeling and Simulation

1. B. Feng
2. Zuo-nong Zhu
3. Liming Ling
4. B. Guo
5. Jianke Yang
6. Deniz Bilman
7. Peter D. Miller
8. Yong Chen

In Statistical and Nonlinear Physics

1. Jianke Yang
2. B. Feng
3. Zuo-nong Zhu
4. Liming Ling
5. Yong Chen
6. Yasuhiro Ohta
7. B. Guo
8. Dmitry A. Zezyulin

Citations in the corpus

• Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations (2013, methodology): “The high-order DT of this type was obtained in reference [8] by the limit technique.”
• Dynamics of rogue waves in the Davey–Stewartson II equation (2012, background): “Ik (Some figures may appear in colour only in the online journal)”
• Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation (2015, methodology): “The spatiotemporal pattern of the second order RW solution for the CSP equation is similar to the ones for the NLS equation [46] or for the derivative NLS equation [47].”
• Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants (2019, background): “The corresponding solution u (19) has two different dynamic behaviors.”
• High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations (2012, methodology): “[12, 13], Ohta and Yang work in the framework of Hirota bilinear method [14] while the present authors use the gDT as a tool [15].”
• Generalized Darboux transformation and localized waves in coupled Hirota equations (2013, methodology): “To overcome this problem, the so-called generalized DT was put forward by Guo, Ling and Liu to investigate rogue wave solutins of the scalar NLS equation [21], then it was soon successfully applied to the Hirota equation [21], the derivative NLS equation [23], the AB model [42] and so on [43].”
• Nonlinear waves in PT -symmetric systems (2016, background): “They 38 can be readily obtained from higher-order rogue waves of the NLS equation through the reduction (109)–(110) (Akhmediev, Ankiewicz, and Soto-Crespo, 2009; Dai and Huang, 2014; Dubard et al., 2010; Guo, Ling, and Liu, 2012; Ohta and Yang, 2012).”
• A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation (2017, methodology): “ImfT 11(l)g+2RefR 0(l)gImfT 12(l)g+ImfT 22(l)g n[22](l; x,t) +[2RefR 0(l)gRefT 11(l)g 2ImfR 0(l)gImfT 11(l)g+2RefT 12(l)g]m[12](l; x,t) + (1 +jR 0(l)j2)RefT 11(l)g 2ImfR 0(l)gImfT 12(l)g+RefT 22(l)g m[22](l; x,t), where we have also used T 21 = T 12, m[jk] = m[kj], and n[jk] = n[kj]. Now combining t”