Exact Solution for Fuzzy-Informed Nonlinear Bi-Hamiltonian Camassa–Holm Model for Shallow-Water Waves

In this paper, the analytical properties of the fuzzy Camassa–Holm equation (FCHE) with a sophisticated model are investigated and used to describe nonlinear wave phenomena under fuzzy conditions. The fuzzy set theory is employed to incorporate uncertainty into physical parameters in shallow water dynamics. The primary objective is to transform the nonlinear FCHE into a manageable ordinary differential equation through a wave transformation variable. The tangent–cotangent (tan–cot) expansion method is developed and applied to derive exact traveling wave solutions by taking into account fuzzy initial conditions. A nonlinear balancing approach is used to derive peakon and periodic solutions. The combined tan–cot method is applied because it can describe wave propagation in both directions. It also provides a useful way to model fluid behavior in turbulent air and offers reasonable predictions.