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On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients
Hamdy I Abdel-Gawad1, Mohamed Osman1
1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
Journal of Advanced Research
|July 23, 2015
Summary
This study examines shallow water waves using the variable coefficient Korteweg-de Vries equation. It reveals wave structures depend on dispersion and nonlinearity, with potential for collapse if unbalanced.
Area of Science:
- Fluid dynamics
- Nonlinear wave phenomena
- Mathematical physics
Background:
- Shallow water waves exhibit complex behaviors.
- The Korteweg-de Vries (KdV) equation models these waves.
- Variable coefficients introduce further complexity.
Purpose of the Study:
- Analyze shallow water wave progression under the variable coefficient KdV (vcKdV) equation.
- Investigate wave behavior based on the proportionality of dispersion and nonlinearity coefficients.
- Develop exact solutions and a novel algorithm for coupled nonlinear partial differential equations.
Main Methods:
- Extended unified method for finding exact solutions.
- Analysis of two cases: proportional and non-linearly dependent dispersion/nonlinearity coefficients.
- Mathematical modeling of water wave dynamics.
Main Results:
- When dispersion and nonlinearity coefficients are proportional, waves exhibit KdV-like geometric structures but with time-dependent speeds.
- Wave structure is maintained only when nonlinearity balances dispersion; otherwise, wave collapse occurs.
- Exact solutions for the vcKdV equation were derived.
Conclusions:
- The balance between nonlinearity and dispersion is critical for maintaining water wave structure.
- The extended unified method provides a powerful tool for solving complex nonlinear PDEs.
- This research offers insights into the dynamics of shallow water waves with variable coefficients.
Keywords:
Jacobi doubly periodic wave solutionsSolitary and periodic wave solutionsThe extended unified methodTime-dependent coefficientsVariable coefficientMore Related Videos
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