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The $b$-family of Camassa-Holm ($b$-CH) equation is a one-parameter family ofPDEs, which includes the completely integrable Camassa-Holm andDegasperis-Procesi equations but possesses different Hamiltonian structures.Motivated by this, we study the existence and the orbital stability of thesmooth solitary wave solutions with nonzero constant background to the $b$-CHequation for the special case $b=1$, whose the Hamiltonian structure isdifferent from that of $b\neq1$. We establish a connection between thestability criterion for the solitary waves and the monotonicity of a singularintegral along the corresponding homoclinic orbits of the spatial ODEs. Weverify the latter analytically using the framework for the monotonicity ofperiod function of planar Hamiltonian systems, which shows that the smoothsolitary waves are orbitally stable. In addition, we find that the existenceand orbital stability results for $0<b<1$ are similar to that of $b>1$,particularly the stability criteria are the same. Finally, combining with theresults for the case $b>1$, we conclude that the solitary waves to the $b$-CHequation is structurally stable with respect to the parameter $b>0$.

The stability of smooth solitary waves for the $b$-family of Camassa-Holm equations