Nonlinear evolution and formation of cosh-Gaussian breathers in cubic-quintic media possessing hump-shaped quintic nonlinearity

We investigate the nonlinear evolution and formation of cosh-Gaussian (ChG) breathers in a nonlinear optical medium characterized by a cubic-quintic (CQ) refractive index profile with a spatially modulated hump-shaped quintic nonlinearity. The study is motivated by the unique intensity structure of ChG beams, which combines hyperbolic cosine and Gaussian features, enabling a rich interplay between diffraction, self-focusing and nonlinear saturation effects. Employing the moment method, we derive a set of evolution equations governing the beam width and axial phase of the laser beam under the influence of the composite nonlinearity. The analysis reveals critical conditions under which the beam undergoes periodic breathing behavior, indicating the formation of robust breather solitons. The presence of the hump-shaped quintic term introduces a localized nonlinear potential that significantly alters the propagation dynamics of the breather.