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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:math> u{u}_{x} . In this brief communication, we show that the mechanical balance underlying this coexistence is disrupted by a stronger nonlinear dissipation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:math> {\left({u}^{2}{u}_{x})}_{x} , which arises from an enhanced buoyancy term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:math> u{u}_{x} . A numerical study is performed to verify the theoretical study.