Pulsating multiplet solutions of quintic wave equations

Numerical studies are used to demonstrate that, in addition to supporting conventional solitons, the quintic Korteweg-de Vries equation, u_t + (u²)_x = u_xxxxx, and its regularized version, u_t+(u²_x+u_txxxx = 0, support multihumped solitary waves (doublets, triplets, quadruplets, etc.), referred to collectively as multiplets. Their peaks pulsate as they travel and undergo nearly elastic collisions with other multiplets. An N-humped multiplet can pulsate thousands of cycles before disassociating into an (N - 1)-humped multiplet and a single-peak solitary wave (singlet). Although multiplets are easily created from an initial wide compact pulse, they rarely are formed by fusing singlets or multiplets in collisions. We describe the emergence and evolution of multiplets, their nearly elastic collision dynamics, and their eventual decomposition into singlets. To consider the effect of cubic dispersion on the solution of these equations, we also study u_t + (u²)_x + ηu_xxx= δu_xxxxx. The impact of cubic dispersion critically depends on the sign of η and its amplitude. For sufficiently large η > η₁ > 0, only a train of singlets emerge from an initial pulse with compact support. If η is decreased, multiplets begin to emerge leading the train of singlets. The number of humps in the multiplet increases as η is decreased, until below a critical point η < η_∝ < 0 the initial pulse decomposes into highly oscillatory waves.