The present article deals with M-soliton solution and N-soliton solution of the (2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long-wave limit. Mainly, by choosing the specific parameter constraints in the M-soliton and N-soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two-, three-, four-, and five-soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.

In this paper, we apply the shooting method for solving the Fuzzy Boundary Value Differential Equations (FBVDEs) of the second order under generalized differentiability. By this method an FBVDE of the second order will be replaced with two fuzzy initial value differential equations and the answers of each of them are obtained by the Adomian method. Finally via linear combination of their solutions, the fuzzy solution will be obtained.

The Hirota bilinear method is employed for searching the localized waves, lump–solitons, and solutions between lumps and rogue waves for the fractional generalized Calogero–Bogoyavlensky–Schiff–Bogoyavlensky–Konopelchenko (CBS-BK) equation. We probe three cases including lump (combination of two positive functions as polynomial), lump–kink (combination of two positive functions as polynomial and exponential function) called the interaction between a lump and one line soliton, and lump–soliton (combination of two positive functions as polynomial and hyperbolic cos function) called the interaction between a lump and two-line solitons. At the critical point, the second-order derivative and the Hessian matrix for only one point will be investigated and the lump solution has one maximum value. The moving path of the lump solution and also the moving velocity and the maximum amplitude will be obtained. The graphs for various fractional orders α are plotted to obtain 3D plot, contour plot, density plot, and 2D plot. The physical phenomena of this obtained lump and its interaction soliton solutions are analyzed and presented in figures by selecting the suitable values. That will be extensively used to report many attractive physical phenomena in the fields of fluid dynamics, classical mechanics, physics, and so on.