Nonlinear Waves and Solitons on Contours and Closed Surfaces

Ludu, Andrei

doi:10.1007/978-3-642-22895-7

Cited by 26 publications

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“…From the Fresnel integrals [25] we have (11) where a is the tension to the bending ratio: γ 0,1/2 /k c . The boundary conditions written in terms of y and x are as follows:…”

Section: Disclination Space Loopmentioning

confidence: 99%

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Theoretical predictions of disclination loop growth for nematic liquid crystals under capillary confinement

et al. 2014

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The combination of low elasticity modulus, anisotropy, and responsiveness to external fields drives the rich variety of experimentally observed pattern formation in nematic liquid crystals under capillary confinement. External fields of interest in technology and fundamental physics are flow fields, electromagnetic fields, and surface fields due to confinement. In this paper we present theoretical and simulation studies of the pattern formation of nematic liquid crystal disclination loops under capillary confinement including branching processes from a m=+1 disclination line to two m=+1/2 disclination curves that describe the postnucleation and growth regime of the textural transformation from radial to planar polar textures. The early postnucleation and growth of emerging disclination loops in cylindrical capillaries are characterized using analytical and computational methods based on the nematic elastica that takes into account line tension and line bending stiffness. Using subdiffusive growth and constant loop anisotropy, we found that the solution to the nematic elastica is a cusped elliptical geometry characterized by exponential curvature variations. The scaling laws that govern the loop growth reflect the tension to bending elasticity balance and reveal that the loop dilation rate depends on the curvature and normal velocity of the disclination. The line energy growth is accommodated by the decrease in branch-point curvature. These findings contribute to the evolving understanding of textural transformations in nematic liquid crystals under confinement using the nematic elastic methodology.

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“…From the Fresnel integrals [25] we have (11) where a is the tension to the bending ratio: γ 0,1/2 /k c . The boundary conditions written in terms of y and x are as follows:…”

Section: Disclination Space Loopmentioning

confidence: 99%

“…where L(t) is the total length of the disclination curve and (t) is the dilation of the loop [25,26]. Hence the timedependent dilation (t) encapsulates the geometry [κ(s,t)] and kinematics [ω n (s,t)] of the growing disclination loop.…”

Section: Kinematic Characterizationmentioning

confidence: 99%

Theoretical predictions of disclination loop growth for nematic liquid crystals under capillary confinement

et al. 2014

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“…In what follows, revising the results presented in [40,41] we will use the travelling wave solutions of the focusing mKdV equation (21) to obtain exact solutions of the curve flow (14) in explicit analytic form. For that purpose, we change to a moving coordinate frame ξ = s − ct, τ = t (22) and assume κ(ξ , τ) =κ(ξ ).…”

Section: Travelling Wave Solutions Of the Mkdv Curve Flowmentioning

confidence: 99%

Traveling Wave Solutions of the Gardner Equation and Motion of Plane Curves Governed by the mKdV Flow

Vassilev¹,

Djondjorov²,

Hadzhilazova³

et al. 2011

AIP Conference Proceedings

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Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers Phys. Fluids 20, 105102 (2008) Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point J. Math. Phys. 48, 065301 (2007) A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations J. Math. Phys. 48, 065204 (2007) The generalized added mass revised Phys. Fluids 19, 058103 (2007) Is plane-channel flow a friendly case for the testing of large-eddy simulation subgrid-scale models?Abstract. The Gardner equation is well-known in the mathematical literature since the late sixties of 20th century. Initially, it appeared in the context of the construction of local conservation laws admitted by the KdV equation. Later on, the Gardner equation was generalized and found to be applicable in various branches of physics (solid-state and plasma physics, fluid dynamics and quantum field theory). In this paper, we examine the travelling wave solutions of the Gardner equation and derive the full set of solutions to the corresponding reduced equation in terms of Weierstrass and Jacobi elliptic functions. Then, we use the travelling wave solutions of the focusing mKdV equation and obtain in explicit analytic form exact solutions of a special type of plane curve flow, known as the mKdV flow.

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“…3.2 in [121]) satisfy the equations In some sense, the higher the dimension of the space, the closer the points are to their topological boundary.…”

Section: Multivariate Networkmentioning

confidence: 99%

Boundaries of a Complex World

Ludu

2016

Springer Series in Synergetics

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Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systemscutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The three major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations, and the "SpringerBriefs in Complexity" which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foundations of the science of complex systems.Through many enduring classic texts, such as Haken's Synergetics and Information and Self-Organization, Gardiner's Handbook of Stochastic Methods, Risken's The Fokker Planck-Equation or Haake's Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field.The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach.More information about this series at

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Nonlinear Waves and Solitons on Contours and Closed Surfaces

Cited by 26 publications

References 0 publications

Theoretical predictions of disclination loop growth for nematic liquid crystals under capillary confinement

Theoretical predictions of disclination loop growth for nematic liquid crystals under capillary confinement

Traveling Wave Solutions of the Gardner Equation and Motion of Plane Curves Governed by the mKdV Flow

Boundaries of a Complex World

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