The zero-dispersion regime of energy-critical fractional nonlinear equations

This work investigates the dispersion limit of the fractional complex Ginzburg–Landau (FCGL) equation in the energy-critical defocusing regime. Using an energy-method framework, we prove that, as the phase parameter θ → 0 , solutions of the FCGL equation with initial data in H α ( R N ) , N = 3 , 4 , converge to solutions of the energy-critical fractional nonlinear heat (FNLH) equation, for which the limiting dynamics are well posed under the stronger regularity assumption H 2 α ( R N ) . The analysis highlights the additional analytical challenges introduced by the nonlocal fractional Laplacian in the energy-critical setting. Numerical simulations are performed using a Fourier spectral discretization in space, which exactly resolves the fractional Laplacian through its Fourier symbol, combined with a fourth-order Runge–Kutta (RK4) scheme in time. Numerical errors between the FCGL and FNLH solutions are computed for different values of θ , and a temporal convergence study is carried out for fixed θ , confirming the consistency and accuracy of the numerical scheme. As θ decreases, the solutions transition from dispersive, oscillatory dynamics to smoother profiles that closely approximate the FNLH solution. Quantitative measures, including the evolution of the L 2 -norm and representative spatial profiles, provide clear numerical confirmation of the theoretical dispersion limit. These results elucidate the interplay between dispersion and dissipation in energy-critical fractional PDEs and contribute to the understanding of nonlocal dynamics in complex media.