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"Elliptic functions."
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The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of
Brunn-Minkowski type for a nonlinear capacity,
In the first part of this article, we prove the Brunn-Minkowski inequality for this
capacity:
In the second part of this article we study a Minkowski problem for a certain measure associated with a compact
convex set
eBook
Elliptic Theory for Sets with Higher Co-dimensional Boundaries
Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher
co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields
a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.
To this end, we turn to degenerate
elliptic equations. Let
In another article to appear, we will prove that when
eBook
Solitary wave solutions for the nonlinear Schrodinger equation with power law nonlinearity
In this paper, the nonlinear Schrödinger equation with power law nonlinearity is considered. This equation admits Lax pair and is integrable. The Jacobi elliptic function approach is used to obtain solitary waves solutions. This method is applied to get the exact solutions for various nonlinear equations. 2D and 3D plots of the obtained solutions are presented. It is discovered that the solutions acquired are crucial for the explanation of a few physical issues.
Journal Article
JACOBIAN ELLIPTIC FUNCTIONS IN SIGNATURE FOUR
The signature four elliptic theory of Ramanujan is provided with a counterpart to the Jacobian modular sine; this counterpart yields natural direct proofs of several hypergeometric identities recorded by Ramanujan, bypassing the signature four transfer principle of Berndt et al. [‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc. 347 (1995), 4163–4244].
Journal Article
He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions
In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.
Journal Article
On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line
This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the
(
G
′
/
G
)
-expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.
Journal Article
Optical solitons of the coupled nonlinear Schrödinger’s equation with spatiotemporal dispersion
In this work, the coupled nonlinear Schrödinger’s equation (CNLSE) is studied with four forms of nonlinearity. The nonlinearities that are considered in this paper are the Kerr law, power law, parabolic law and dual-power law. Jacobi elliptic function solutions and also bright and dark optical soliton solutions are obtained for each law of the CNLSE. We will acquire constraint conditions for the existence of obtained solitons.
Journal Article
Optical solitons in parabolic law medium: Jacobi elliptic function solution
The aim of this paper is to obtain optical soliton solutions by nonlinear dispersion in parabolic law medium by using Jacobi elliptic functions. The presented problem is studied with Kerr law nonlinearity, and dark and bright solitons are acquired. Also, we develop the stability analysis for problem.
Journal Article
Reformulation of the standard theory of Fowler–Nordheim tunnelling and cold field electron emission
This paper presents a major reformulation of the standard theory of Fowler–Nordheim (FN) tunnelling and cold field electron
emission (CFE). Mathematical analysis and physical interpretation become easier if the principal field emission elliptic function
Journal Article