Persistence of a unique periodic wave train in convecting shallow water fluid

Abstract

The coexistence of a traveling pulse and a periodic traveling wave was established in a convecting shallow water model when taking a nonlinear buoyancy term uuxu{u}_{x}. In this brief communication, we show that the mechanical balance underlying this coexistence is disrupted by a stronger nonlinear dissipation (u2ux)x{\left({u}^{2}{u}_{x})}_{x}, which arises from an enhanced buoyancy term u2ux{u}^{2}{u}_{x}. Consequently, the convecting shallow water model exhibits either a unique periodic wave or a unique solitary wave, each within a fixed range of wave speeds. Furthermore, we show that the wave speed is monotonic with respect to the wave amplitude and is smaller than that observed in the model with the buoyancy term uuxu{u}_{x}. A numerical study is performed to verify the theoretical study.

Keywords

• abelian integral
• solitary wave
• periodic traveling wave
• 35b10
• 35b20
• 35b25