Multi-hump stationary waves for a Korteweg–de Vries equation with nonlocal perturbations

Periodic and solitary wave solutions are investigated numerically for a perturbed Korteweg–de Vries equation with unstable and dissipation terms in Hilbert transform: ut+uux+uxxx+η(Hux+Huxxx)=0. A family of solitary wave solutions S(1),S(2),…,S(n),…, whose members are distinguished by the number of “humps”, is numerically identified. The tails of these waves decay as O(1/∣x∣2) when ∣x∣→∞ irrespective of the magnitude of η. It is also found that for a given η, there exist families of periodic wave solutions P(1),P(2),…,P(n),…, which originate from one near-sinusoidal wave and end up in the infinite periodicity to the corresponding solitary waves. The numerical results are consistent with the theoretical estimates based on the conservation properties.