Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms
In this paper, a generalized Benjamin-Bona-Mahony (BBM) equation with two different types of weak dissipative effects is studies, \[(u^3)_t + (u^4)_x + u_{xxx}+\epsilon P_i = 0, \,\,\,\, i=1,\,2. \] The term \(P_1\) describes the weak second and fourth derivative diffusions without Marangoni effect, \[ P_1 = u_{xx} + u_{xxxx}.\] The term \(P_2\), on the other hand, describes a generalized Marangoni effect, \[ P_2 = ((a_0 + a_1u + a_2u^2)u_x)_x, \] where \(a_0\), \( a_1\) and \(a_2\) are bounded parameters. The equation is analyzed through a reduction to the case of \(\epsilon=0\). The paper is devoted to the study of solitary and periodic waves in these two cases, including the issues of existence of the waves, uniqueness, co-existence of solitary and periodic waves, and monotonicity. The proofs are based on the theory of weak Hilbert's 16th problem and bifurcation theory.
generalized Benjamin-Bona-Mahony equation
scientific article; zbMATH DE number 7000401
scientific article