TY - JOUR

T1 - Drops and Fingers in a Tempered Ginzburg-Landau set-up

AU - Rosenau, Philip

N1 - Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/11

Y1 - 2021/11

N2 - We introduce gradient-tempered Ginzburg-Landau, GL, free energy functional J=∫(G(ux)−W(u))dx, where W(u)=λ2(2u2−u4) is the bulk energy endowed with a stiffness coefficient a2(u) which vanishes both at the phases and on the interface: a2(u)=|u|2γ|1−u2|2β, [Formula presented]<β,γ≤1. It induces compact drops and interphase transitions within a finite domain. The assumed interface energy G=1+a2ux2−1, tempers the divergence of gradients in the ultra-violet limit. Consequently, when λ2, 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, ut=−[Formula presented]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.

AB - We introduce gradient-tempered Ginzburg-Landau, GL, free energy functional J=∫(G(ux)−W(u))dx, where W(u)=λ2(2u2−u4) is the bulk energy endowed with a stiffness coefficient a2(u) which vanishes both at the phases and on the interface: a2(u)=|u|2γ|1−u2|2β, [Formula presented]<β,γ≤1. It induces compact drops and interphase transitions within a finite domain. The assumed interface energy G=1+a2ux2−1, tempers the divergence of gradients in the ultra-violet limit. Consequently, when λ2, 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, ut=−[Formula presented]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.

KW - Drops

KW - Fingers

KW - Tempered Ginzburg–Landau

UR - http://www.scopus.com/inward/record.url?scp=85109474920&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2021.132956

DO - 10.1016/j.physd.2021.132956

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AN - SCOPUS:85109474920

SN - 0167-2789

VL - 425

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

M1 - 132956

ER -