Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics
• Wave Phenomena

Background:

• Dispersionless Whitham modulation equations in 2+1 dimensions describe wave behavior.
• Identifying instabilities is crucial for understanding wave dynamics.
• Lighthill instability surfaces mark critical thresholds in wave propagation.

Purpose of the Study:

• To review and identify instabilities in 2+1 dispersionless Whitham modulation equations.
• To reformulate modulation theory near Lighthill instability thresholds.
• To explore the properties and applications of the resulting nonlinear phase modulation equation.

Main Methods:

• Review of dispersionless Whitham modulation equations in 2+1 dimensions.
• Reformulation of modulation theory near Lighthill instability thresholds using slow phase, moving frame, and specific scalings.
• Application of the derived nonlinear phase modulation equation to a complex nonlinear 2+1 Klein-Gordon equation.

Main Results:

• Identification of instabilities in the 2+1 dispersionless Whitham modulation equations.
• Derivation of a nonlinear phase modulation equation, a geometric form of the 2+1 two-way Boussinesq equation, near Lighthill surfaces.
• Demonstration that this Boussinesq equation is universal, dispersive, and admits multi-periodic, quasi-periodic, and multi-pulse localized solutions.
• Application to a Klein-Gordon equation revealing two Lighthill surfaces.

Conclusions:

• The reformulated modulation theory provides a universal and dispersive 2+1 two-way Boussinesq equation.
• This equation exhibits rich localized wave solutions, applicable to complex systems like the Klein-Gordon equation.
• The findings contribute to the understanding of nonlinear wave stability and patterns.