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Elliptic-Functions

Mathematical research with nonlinear coupled differential equations and elliptic functions. Uses Python, Jupyter, sympy, and mpmath.

This work is interested in:

Solving nonlinear coupled differential equations analytically, often in terms of elliptic functions
Establishing new ways to analyse associated concepts in physics and to obtain new physical insights from the mathematics
Discovering new methods and techniques for solving nonlinear coupled mode differential equations that are generalisable to other physical models
Researching new ideas in symmetry (in particular Parity Time PT symmetry), conservation laws, and Lagrangian and Hamiltonian mechanics
Exploring how to approach such research using open source tools such as Python, sympy and mpmath

Relevant work by the author:

General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

Nonlinear Effects in Multimode Optical Fibers (Chapter 4 in particular)

PT Symmetric Optical Lattices

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Elliptic-Functions

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Name		Name	Last commit message	Last commit date
Latest commit History 39 Commits
LaTex		LaTex
numerical_evaluation		numerical_evaluation
.DS_Store		.DS_Store
.gitignore		.gitignore
Algorithm for higher order derivatives of the Weierstrass P function.ipynb		Algorithm for higher order derivatives of the Weierstrass P function.ipynb
Algorithm for higher order derivatives of the Weierstrass sigma function.ipynb		Algorithm for higher order derivatives of the Weierstrass sigma function.ipynb
Analytic Solutions of Nonlinear Coupled Mode Optics with Quadratic Nonlinearity.ipynb		Analytic Solutions of Nonlinear Coupled Mode Optics with Quadratic Nonlinearity.ipynb
Building Hamiltonians from the Frobenius Stickelberger Elliptic Function Determinant Formula.ipynb		Building Hamiltonians from the Frobenius Stickelberger Elliptic Function Determinant Formula.ipynb
Cross product representations of 2 mode systems.ipynb		Cross product representations of 2 mode systems.ipynb
Cubic Nonlinearity in PT Symmetric Optical Structures.ipynb		Cubic Nonlinearity in PT Symmetric Optical Structures.ipynb
Degenerate determinant formulas for Kronecker theta functions.ipynb		Degenerate determinant formulas for Kronecker theta functions.ipynb
Deriving the Coupled Ordinary Differential Equations of the Weierstrass Sigma Ratios from the Frobenius Stickelberger Elliptic Function Determinant Formula.ipynb		Deriving the Coupled Ordinary Differential Equations of the Weierstrass Sigma Ratios from the Frobenius Stickelberger Elliptic Function Determinant Formula.ipynb
Double sum from Mathstack Exchange as a Kronecker theta function.ipynb		Double sum from Mathstack Exchange as a Kronecker theta function.ipynb
Fifth order Dixon functions as Kronecker theta functions derived from the Frobenius-Stickelberger determinant.ipynb		Fifth order Dixon functions as Kronecker theta functions derived from the Frobenius-Stickelberger determinant.ipynb
Fourth order Dixon functions as Kronecker theta functions derived from the Frobenius-Stickelberger determinant.ipynb		Fourth order Dixon functions as Kronecker theta functions derived from the Frobenius-Stickelberger determinant.ipynb
Functional Relations of the Weierstrass Sigma Solutions.ipynb		Functional Relations of the Weierstrass Sigma Solutions.ipynb
Hamiltonian Systems with Quadratic Forms.ipynb		Hamiltonian Systems with Quadratic Forms.ipynb
Hamiltonian for 3 modes from FS determinant.ipynb		Hamiltonian for 3 modes from FS determinant.ipynb
Hamiltonians from products of Kronecker theta functions.ipynb		Hamiltonians from products of Kronecker theta functions.ipynb
Jacobi Elliptic Functions as Kronecker Theta Functions.ipynb		Jacobi Elliptic Functions as Kronecker Theta Functions.ipynb
Kronecker Theta Functions and Kronecker Double Series.ipynb		Kronecker Theta Functions and Kronecker Double Series.ipynb
Lagrangian with 6 modes.ipynb		Lagrangian with 6 modes.ipynb
Lagrangians.ipynb		Lagrangians.ipynb
Log Hamiltonians.ipynb		Log Hamiltonians.ipynb
Lotka–Volterra Like Systems With Generalised Exponents.ipynb		Lotka–Volterra Like Systems With Generalised Exponents.ipynb
Lotka–Volterra Like Systems.ipynb		Lotka–Volterra Like Systems.ipynb
Mobius Transforming to Higher Order Coupled Mode Systems.ipynb		Mobius Transforming to Higher Order Coupled Mode Systems.ipynb
On the connection between Dixon Functions and Kronecker Theta Functions-Copy1.ipynb		On the connection between Dixon Functions and Kronecker Theta Functions-Copy1.ipynb
On the connection between Dixon Functions and Kronecker Theta Functions.ipynb		On the connection between Dixon Functions and Kronecker Theta Functions.ipynb
On the integral representation of the Kronecker theta function.ipynb		On the integral representation of the Kronecker theta function.ipynb
P and Q Hamiltonians with Kronecker theta solutions.ipynb		P and Q Hamiltonians with Kronecker theta solutions.ipynb
README.md		README.md
Relating Carlson Symmetric Elliptic Integrals to the Weierstrass Elliptic Functions.ipynb		Relating Carlson Symmetric Elliptic Integrals to the Weierstrass Elliptic Functions.ipynb
Relating the Kronecker theta and Weierstrass P functions.ipynb		Relating the Kronecker theta and Weierstrass P functions.ipynb
Series expansions of the Kronecker theta function.ipynb		Series expansions of the Kronecker theta function.ipynb
Studying 2 modes systems through quadratic forms - Reducing parameters.ipynb		Studying 2 modes systems through quadratic forms - Reducing parameters.ipynb
Studying 2 modes systems through quadratic forms.ipynb		Studying 2 modes systems through quadratic forms.ipynb
The Kronecker theta model with 5 modes.ipynb		The Kronecker theta model with 5 modes.ipynb
Weierstrass Sigma Ratios from the Frobenius Stickelberger Elliptic Function Determinant Formula - With Q and P Polynomials.ipynb		Weierstrass Sigma Ratios from the Frobenius Stickelberger Elliptic Function Determinant Formula - With Q and P Polynomials.ipynb
nth_diff_pw.py		nth_diff_pw.py
requirements.txt		requirements.txt
wpk.py		wpk.py

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