<i>The Calculus of Variations</i>

Akhiezer, N. I.; Frink, Aline H.; Teichmann, T.

doi:10.1063/1.3050816

Cited by 41 publications

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“…Taking into account that only stable phase configurations, corresponding to minima of (3), are physically observable, the problem one has to solve can be formulated as follows: Find all solutions to (7), (8) that minimize (at least locally) (3) at given H ≥ 0. To resolve this problem, we have to turn to sufficient conditions of the minimum, 18 which requires an analysis of the second variation of (3).…”

Section: Introductionmentioning

confidence: 99%

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Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

Kuplevakhsky

Glukhov

2006

Phys. Rev. B

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The problem of vortex structure in a single Josephson junction in an external magnetic field, in the absence of transport currents, is reconsidered from a new mathematical point of view. In particular, we derive a complete set of exact analytical solutions representing all the stationary points (minima and saddle-points) of the relevant Gibbs free-energy functional. The type of these solutions is determined by explicit evaluation of the second variation of the Gibbs free-energy functional. The stable (physical) solutions minimizing the Gibbs free-energy functional form an infinite set and are labelled by a topological number Nv = 0, 1, 2, . . .. Mathematically, they can be interpreted as nontrivial "vacuum" (Nv = 0) and static topological solitons (Nv = 1, 2, . . .) of the sine-Gordon equation for the phase difference in a finite spatial interval: solutions of this kind were not considered in previous literature. Physically, they represent the Meissner state (Nv = 0) and Josephson vortices (Nv = 1, 2, . . .). Major properties of the new physical solutions are thoroughly discussed. An exact, closed-form analytical expression for the Gibbs free energy is derived and analyzed numerically. Unstable (saddle-point) solutions are also classified and discussed.

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Section: Introductionmentioning

confidence: 99%

“…The main property of the functional (3) follows from the fact that it belongs to the class of regular functionals, i.e., satisfies a necessary condition of the minimum; 18 hence, all stationary points of (3) are either minima of saddle points. [Unfortunately, the statement of Ref.…”

Section: Introductionmentioning

confidence: 99%

Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

Kuplevakhsky

Glukhov

2006

Phys. Rev. B

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“…However, contrary to popular belief, many other physical laws do not have a mechanistic 500 SASCHA E. ENGELBRECHT 2 An introduction to the calculus of variations can be found in Dreyfus (1965), and in-dept treatments are provided by Akhiezer (1962) and Bliss (1946). For a history of the calculus of variations, see Goldstine (1980).…”

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confidence: 99%

Minimum Principles in Motor Control

Engelbrecht¹

2001

Journal of Mathematical Psychology

143

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“…To eliminate any questions about the actual equilibrium field configurations in layered superconductors at H > 0, we present an explicit mathematical proof that the Meissner (0-soliton) and vortex-plane (soliton) configurations are unique solutions that satisfy the conditions of the minimum (1), (2). Moreover, we show that all the minima are strict and strong [19]. For the sake of diversification, we employ a new method of exact minimization of the Gibbs free-energy functional that, in contrast to Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Mathematically, a classification of stable solutions amounts to the determination of all points of local minima of the energy functionals. A local minimum of the Gibbs free-energy functional of layered superconductors is determined by the relations [18,19] …”

Section: Introductionmentioning

confidence: 99%

Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

Kuplevakhsky

2004

Low Temperature Physics

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We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H > 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizes of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, non-soliton configurations (interpreted in some previous publications as «isolated fluxons» and «fluxon lattices») are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and establish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are derived: They are nothing but the «vacuum state» and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0 £ < ¥ H and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H to be sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics.

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The Calculus of Variations

Cited by 41 publications

References 0 publications

Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

Static solitons of the sine-Gordon equation and equilibrium vortex structure in Josephson junctions

Minimum Principles in Motor Control

Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

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