The dynamics of geometric PDEs: Surface evolution equations and a comparison with their small gradient approximations

2019 ◽  
Vol 29 (10) ◽  
pp. 103119
Author(s):  
C. Kabelitz ◽  
S. J. Linz
Keyword(s):  
Evolution Equations ◽  
Surface Evolution ◽  
Geometric Pdes

scholarly journals An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity

1997 ◽  
Vol 38 (4) ◽  
pp. 518-541 ◽  
Author(s):  
Peter Tritscher
Keyword(s):  
Nonlinear Equation ◽  
Surface Diffusion ◽  
Anisotropic Material ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Constitutive Relations ◽  
Nonlinear Evolution Equations ◽  
Surface Evolution ◽  
Integrable Nonlinear Equation ◽  
Closed Form Solutions

AbstractMembers of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

Formation of Regular Domain Structures in Quenched Ferroelectrics Under the Influence of an External High-frequency Electric Field

2019 ◽  
Vol 9 (3) ◽  
pp. 344-352 ◽  
Author(s):  
L.I. Stefanovich ◽  
O.Y. Mazur ◽  
V.V. Sobolev
Keyword(s):  
Lithium Niobate ◽  
Electric Field ◽  
Domain Structure ◽  
High Frequency ◽  
Evolution Equations ◽  
Regular Domain ◽  
Field Frequency ◽  
Domain Structures ◽  
Alternating Electric Field ◽  
Lithium Niobate Crystals

Introduction: Within the framework of the phenomenological theory of phase transitions of the second kind of Ginzburg-Landau, the kinetics of ordering of a rapidly quenched highly nonequilibrium domain structure is considered using the lithium tantalate and lithium niobate crystals as an example. Experimental: Using the statistical approach, evolution equations describing the formation of the domain structure under the influence of a high-frequency alternating electric field in the form of a standing wave were obtained. Numerical analysis has shown the possibility of forming thermodynamically stable mono- and polydomain structures. It turned out that the process of relaxation of the system to the state of thermodynamic equilibrium can proceed directly or with the formation of intermediate quasi-stationary polydomain asymmetric phases. Results: It is shown that the formation of Regular Domain Structures (RDS) is of a threshold character and occurs under the influence of an alternating electric field with an amplitude less than the critical value, whose value depends on the field frequency. The conditions for the formation of RDSs with a micrometer spatial scale were determined. Conclusion: As shown by numerical studies, the RDSs obtained retain their stability, i.e. do not disappear even after turning off the external electric field. Qualitative analysis using lithium niobate crystals as an example has shown the possibility of RDSs formation in high-frequency fields with small amplitude under resonance conditions

Sign in / Sign up

Export Citation Format

Share Document