Drops and Fingers in a Tempered Ginzburg-Landau set-up

Abstract We introduce gradient-tempered Ginzburg-Landau, GL, free energy functional J = ∫ ( G ( u x ) − W ( u ) ) d x , where W ( u ) = λ 2 ( 2 u 2 − u 4 ) is the bulk energy endowed with a stiffness coefficient a 2 ( u ) which vanishes both at the phases and on the interface: a 2 ( u ) = | u | 2 γ | 1 − u 2 | 2 β , 1 2 β , γ ≤ 1 . It induces compact drops and interphase transitions within a finite domain. The assumed interface energy G = 1 + a 2 u x 2 − 1 , tempers the divergence of gradients in the ultra-violet limit. Consequently, when λ 2 , the bulk energy strength parameter crosses a critical value, depending on their amplitude, the forming drops may develop at their edges sharp jumps and turn into fingers across which the energy remains finite. In the Tempered Allen-Cahn, u t = − δ δ x J equation, the resulting flux-saturating diffusion delays the resolution of initial discontinuities and if 1 ≤ γ , it blocks excitation’s spread beyond its initial span. A number of explicit spherical solutions is also presented.