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Analytical and numerical studies for integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies.

Chaos (Woodbury, N.Y.)·2025

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Updated: Jun 11, 2025

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

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Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation.

Ya-Hui Huang¹, Rui Guo¹

¹School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China.

Chaos (Woodbury, N.Y.)

|October 1, 2024

Summary

This study investigates wave breaking using the complex modified KdV equation, revealing that dispersive regularization creates dispersive shock waves (DSW). The research accurately predicts DSW properties, including phase shifts, matching numerical simulations.

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Area of Science:

Nonlinear Physics
Fluid Dynamics
Wave Phenomena

Background:

Wave breaking is a critical phenomenon in nonlinear systems.
The complex modified KdV (cmKdV) equation models certain wave behaviors.
Dispersive regularization offers a way to study singularities in wave equations.

Purpose of the Study:

To analyze wave breaking in the defocusing cmKdV equation.
To describe the formation and properties of dispersive shock waves (DSW).
To provide a complete analytical description of DSW, including phase shifts.

Main Methods:

Utilizing the Gurevich-Pitaevskii approach and Whitham modulation theory.
Applying the generalized hodograph method to solve Whitham equations.
Deriving generalized phase relationships and modified Gurevich-Pitaevskii matching conditions.

Main Results:

Wave breaking in the cmKdV equation leads to the generation of a DSW.
The DSW is characterized as a modulated periodic wave.
Accurate determination of DSW boundaries and phase shifts was achieved.
Analytical predictions show excellent agreement with numerical simulations.

Conclusions:

The study successfully describes DSW formation and properties from wave breaking.
The generalized phase relationships and matching conditions provide a complete DSW description.
The findings validate the theoretical framework against numerical evidence.