Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified Korteweg-de Vries equation.

*(English)*Zbl 0985.35077Summary: The Korteweg-de Vries (KdV) equation was derived as a model for weakly nonlinear long waves propagating down a channel when cross-channel and depth variations are sufficiently weak. In this article, we study the steadily translating coherent structures of a generalization of this equation, the rotation-modified Korteweg-de Vries (RMKdV) equation, which applies when Coriolis forces are significant:
\[
(u_t+ uu_x+ u_{xxx})_x= \varepsilon^2 u=0,
\]
where \(x\) is the down-channel coordinate. (This is also called the “Ostrovsky” equation.) The RMKdV solitary waves are weakly nonlocal due to the radiation of long waves where the radiation is proportional to the small parameter \(\varepsilon\).

Because of its simplicity, the RMKdV coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter (micropterons). We extend the matched asymptotic expansions of Hunter to third order. We also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem. New modes, with multiple KdV-like peaks flanked by small quasi-sinusoidal oscillations, are shown to be well described by matched asymptotics, too.

Because of its simplicity, the RMKdV coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter (micropterons). We extend the matched asymptotic expansions of Hunter to third order. We also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem. New modes, with multiple KdV-like peaks flanked by small quasi-sinusoidal oscillations, are shown to be well described by matched asymptotics, too.