Singular solutions of the BBM equation : analytical and numerical study

Sergey Gavrilyuk (1, 2) , Keh-Ming Shyue (3)

1 AMU - Aix Marseille Université
2 IUSTI - Institut universitaire des systèmes thermiques industriels
3 NTU - National Taiwan University [Taiwan]

We show that the Benjamin-Bona-Mahony (BBM) equation admits stable travelling wave solutions representing a sharp transition front linking a constant state with a periodic wave train. The constant state is determined by the parameters of the periodic wave train : the wave length, amplitude and phase velocity, and satisfies both the Rankine-Hugoniot conditions for the corresponding Whitham modulation system and generalized Rankine-Hugoniot conditions for the exact BBM equation. Such stable shock-like travelling structures exist if the phase velocity of the periodic wave train is not less than the solution average value. To validate the accuracy of the numerical method, we derive the (singular) solitary limit of the Whitham system for the BBM equation and compare the corresponding numerical and analytical solutions. We find good agreement between analytical results and numerical solutions.

Mots clés: nonlinear dispersive equations; shock-like transition fronts; Whitham's modulation equations; solitary limit

Domaines: Mathématiques [math]; Science non linéaire [physics]

Sergey Gavrilyuk, Keh-Ming Shyue. Singular solutions of the BBM equation : analytical and numerical study. 2021. ⟨hal-03220465⟩