A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained.
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http://dx.doi.org/10.1103/PhysRevE.79.056602 | DOI Listing |
Phys Rev E
June 2024
School of Mathematics, South China University of Technology, Guangzhou 510640, China.
We present elliptic-rogue wave solutions for integrable nonlinear soliton equations in rational form by elliptic functions. Unlike solutions generated on the plane wave background, these solutions depict rogue waves emerging on elliptic function backgrounds. By refining the modified squared wave function method in tandem with the Darboux-Bäcklund transformation, we establish a quantitative correspondence between elliptic-rogue waves and the modulational instability.
View Article and Find Full Text PDFChaos
May 2024
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, People's Republic of China.
A Kundu-nonlinear Schrödinger equation that can be utilized to simulate the pulse propagation in optical fibers is researched in this paper. First, the Lax integrability of the above equation is proved and its modulational instability (i.e.
View Article and Find Full Text PDFPhys Rev E
April 2024
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China.
Under investigation are the three-component Gross-Pitaevskii equations in F=1 spinor Bose-Einstein condensates. Various localized waves' generation mechanisms have been derived from plane wave solutions using modulation instability. The perturbed continuous waves produce a large number of rogue wave structures through the split-step Fourier numerical method.
View Article and Find Full Text PDFPhys Rev Lett
March 2024
Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, United Kingdom.
The optical Darboux transformer for solitons is introduced as a photonic device that performs the Darboux transformation directly in the optical domain. This enables two major advances for optical signal processing based on the nonlinear Fourier transform: (i) the multiplexing of solitonic waveforms corresponding to different discrete eigenvalues of the Zakharov-Shabat system, and (ii) the selective filtering of an arbitrary number of individual solitons too. The optical Darboux transformer can be built using existing commercially available photonic technology components and constitutes a universal tool for signal processing, optical communications, optical rogue waves generation, and waveform shaping and control in the nonlinear Fourier domain.
View Article and Find Full Text PDFChaos
April 2024
School of Mathematics, South China University of Technology, Guangzhou 510641, China.
This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root.
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