Wave–wave interaction occurs in the propagation deformation of nonlinear long waves in shallow-water. In order to further study the propagation mechanism of shallow-water long waves interaction, the exact traveling wave solutions of the local fractional generalized Hirota–Satsuma coupled Korteweg–de Vries (HS-KdV) equations defined by the Cantor sets are obtained. The non-differentiable solutions with fixed fractal dimension and different propagation velocity are discussed. The results indicate that the exact solutions of the local fractional generalized HS-KdV equations characterize the interaction of fractal long waves with different dispersion relations on shallow-water surfaces.

浅水中非线性长波的传播变形中发生波与波的相互作用。为了进一步研究浅水长波相互作用的传播机制，获得了由Cantor集定义的局部分数阶广义Hirota-Satsuma耦合Korteweg-de Vries (HS-KdV)方程的精确行波解。讨论了具有固定分形维数和不同传播速度的不可微解。结果表明，局部分数阶广义HS-KdV方程的精确解表征了浅水表面不同色散关系的分形长波的相互作用。