The variable separation approach is an important tool to obtain exact localized and fractal structure solutions of higher dimensional nonlinear problems. The general separable solutions involving double random functions are obtained by means of Riccati equation and the variable separation hypothesis. By choosing the suitable arbitrary functions, we obtain a class of localized excitations and fractal structures solutions of the (2 + 1)-dimensional modified dispersive long wave (MDLW) model. Such solutions present couple lump waves, breather wave solutions, dromions, oscillating multi-lumps, oscillating multi-dromoins and ring solitons of the model. Moreover, the fractal lump and fractal dromions type excitation-localized structures of the model are presented here. To visualize the dynamics of each excitation structures are demonstrated by the 3D and density plots.

中文翻译：

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（2 +1）维修正色散长波模型的一类局部孤子和分形图案解

变量分离方法是获取高维非线性问题的精确局部和分形结构解的重要工具。利用Riccati方程和变量分离假设，可以得到涉及双随机函数的一般可分离解。通过选择合适的任意函数，我们获得了（2 +1）维修正色散长波（MDLW）模型的一类局部激发和分形结构解。这样的解决方案呈现出模型的成对的团块波，通气波解决方案，dromions，振荡的多团块，振荡的多dromoins和环形孤子。此外，在此还介绍了模型的分形块和分形dromions型激发局部结构。