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Multiple Lax integrable higher dimensional AKNS(-1) equations and sine-Gordon equations.
Xueping Cheng1, Guiming Jin2, Jianan Wang2
1School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China.
Chaos (Woodbury, N.Y.)
|October 1, 2024
Summary
Researchers extended the AKNS(-1) equations to a (4+1)-dimensional system and generalized the sine-Gordon equation to (3+1) dimensions. Lax integrability was proven, and traveling wave solutions were found for these nonlinear systems.
Area of Science:
- Nonlinear Partial Differential Equations
- Mathematical Physics
- Integrable Systems
Background:
- The AKNS(-1) equations and sine-Gordon equation are fundamental in studying nonlinear phenomena.
- Extending these models to higher dimensions can reveal new complex behaviors.
Purpose of the Study:
- To generalize the (1+1)-dimensional AKNS(-1) and sine-Gordon equations to higher dimensions.
- To investigate the Lax integrability of these extended systems.
- To find traveling wave solutions for the new models.
Main Methods:
- Modified deformation algorithm based on conservation laws.
- Dependent variable transformation.
- Introduction of deformation operators to Lax pairs.
Main Results:
- A (4+1)-dimensional AKNS(-1) system and its degenerated lower-dimensional versions were constructed.
- A (3+1)-dimensional sine-Gordon equation was derived from the (1+1)-dimensional version.
- Lax integrability was proven for both the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation.
- Traveling wave solutions were obtained using tanh functions and incomplete elliptic integrals.
Conclusions:
- The study successfully generalized important nonlinear systems to higher dimensions.
- The proven Lax integrability facilitates further analytical and numerical studies.
- The obtained solutions offer insights into complex physical phenomena described by these nonlinear systems.
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