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Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background
1College of Science, University of Shanghai for Science and Technology, P. O. Box 253, Shanghai 200093, China.
Chaos (Woodbury, N.Y.)
|March 3, 2021
Summary
This study constructs rogue wave solutions for the fourth-order nonlinear Schrödinger (NLS) equation on a periodic background. These solutions exhibit analogs to the standard NLS equation and can form bound states under specific conditions.
Area of Science:
- Nonlinear Optics
- Mathematical Physics
Background:
- The standard nonlinear Schrödinger (NLS) equation describes various phenomena in optics and physics.
- Fourth-order NLS equations introduce higher-order effects, leading to complex wave dynamics.
- Periodic waves, like Jacobi elliptic function solutions (dn- and cn-), are fundamental states in nonlinear systems.
Purpose of the Study:
- To construct rogue wave solutions on a periodic background for the fourth-order NLS equation.
- To investigate the dynamics and properties of these novel rogue wave solutions.
- To explore the relationship between rogue waves and soliton solutions in this model.
Main Methods:
- Construction of dn- and cn-periodic wave solutions.
- Application of nonlinearization of spectral problem combined with Darboux transformation.
- Analysis of rogue wave dynamics and limiting cases.
Main Results:
- Both dn- and cn-periodic waves are modulationally unstable.
- Rogue wave solutions are successfully derived on these unstable periodic backgrounds.
- Rogue waves show analogs to the standard NLS equation; higher-order effects do not alter the magnification factor.
- As the elliptic modulus approaches 1, rogue waves reduce to multi-soliton bound states.
Conclusions:
- The study successfully generates rogue wave solutions for the fourth-order NLS equation on periodic backgrounds.
- The derived rogue waves share dynamic similarities with those in the standard NLS equation.
- The findings offer insights into the complex wave phenomena and soliton interactions in higher-order nonlinear systems.
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