Existence and Orbital Stability of Solitary-Wave Solutions for Higher-Order BBM Equations (Journal of Mathematical Study, 2022)
This paper discusses the existence and stability of solitary-wave solutionsof a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involvespseudo-differential operators for the linear part. One of such equations can be derivedfrom water-wave problems as second-order approximate equations from fully nonlineargoverning equations. Under some conditions on the symbols of pseudo-differentialoperators and the nonlinear terms, it is shown that the general higher-order BBM equationhas solitary-wave solutions. Moreover, under slightly more restrictive conditions,the set of solitary-wave solutions is orbitally stable. Here, the equation has a nonlinearpart involving the polynomials of solution and its derivatives with different degrees(not homogeneous), which has not been studied before. Numerical stability and instabilityof solitary-wave solutions for some special fifth-order BBM equations are alsogiven.
Keywords: Higher-order BBM equations; solitary-wave solutions; orbital stability.