Photonic time-stretch techniques and their applications are reviewed. The approach enables the observation of signals that are otherwise too short or rapid for conventional measurement. Observing non-repetitive and statistically rare signals that occur on short timescales requires fast real-time measurements that exceed the speed, precision and record length of conventional digitizers. Photonic time stretch is a data acquisition method that overcomes the speed limitations of electronic digitizers and enables continuous ultrafast single-shot spectroscopy, imaging, reflectometry, terahertz and other measurements at refresh rates reaching billions of frames per second with non-stop recording spanning trillions of consecutive frames. The technology has opened a new frontier in measurement science unveiling transient phenomena in nonlinear dynamics such as optical rogue waves and soliton molecules, and in relativistic electron bunching. It has also created a new class of instruments that have been integrated with artificial intelligence for sensing and biomedical diagnostics. We review the fundamental principles and applications of this emerging field for continuous phase and amplitude characterization at extremely high repetition rates via time-stretch spectral interferometry.

Time stretch and its applications

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the new Hamiltonian amplitude equation introduced by Wadati et al. When the modulus nt approaches to 1 and 0, then the hyperbolic function solutions (including the solitary wave solutions) and trigonometric function solutions are also given respectively. As the parameter s goes to zero, the new Hamiltonian amplitude equation becomes the well-known nonlinear Schrodinger equation (NLS), and at least there are 37 kinds of solutions of NLS can be derived from the solutions of the new Hamiltonian amplitude equation. (c) 2004 Elsevier Ltd. All rights reserved.

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

https://theory.sinp.msu.ru/~tarasov/PDF/AP2008.pdf

Fractional vector calculus and fractional Maxwell’s equations

Our extended Jacobi elliptic function expansion method is further improved to be a more powerful method, which is still called the extended Jacobi elliptic function expansion method, by using 12 Jacobi elliptic functions. The new (2+1)-dimensional integrable Davey–Stewartson-type is chosen to illustrate the approach. As a consequence, 24 families of Jacobi elliptic function solutions are obtained. When the modulus m→1, these doubly periodic solutions degenerate as soliton solutions. The method can be also applied to other nonlinear differential equations.

Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method

Vector financial rogue waves

We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black—Scholes model. These rogue wave solutions may he used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

https://iopscience.iop.org/article/10.1088/0253-6102/54/5/31/pdf

Financial Rogue Waves

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by using our extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and another three families of new doubly periodic solutions (Jacobian elliptic function solutions) are found again by using a new transformation, which and our extended Jacobian elliptic function expansion method form a new method still called the extended Jacobian elliptic function expansion method. The new method can be more powerful to be applied to other nonlinear differential equations.

http://iopscience.iop.org/article/10.1088/0253-6102/38/2/143/pdf

New Jacobian Elliptic Function Solutions of Modified KdV Equation: II

In this paper, with the help of Mathematica, three travelling wave solutions for a class of nonlinear evolution equations u tt +au xx +bu+cu 3 =0 are obtained by trigonometric function method and Wu-eliminition method which include new trarelling wave solutions, bell soliton solutions and kink soliton solutions Some equations such as Duffing equation, Klein-Gordon equation, Landau-Ginburg-Higgs equation and  4 equation are particular cases of the evolution equations The method also can be applied to other nonlinear equations

New explicit and travelling wave solutions for a class of nonlinear evolution equations

In this paper, we study the generalized coupled Hirota-Satsuma KdV system by using the new generalized transformation in homogeneous balance method. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions, and the combined formal solitary wave solutions, and periodic wave solutions, are obtained

https://faculty.ecnu.edu.cn/picture/article/202/9b/c5/6fcf99134d43ac44fac64f06b611/b267f7d7-4ee9-4fcd-91be-dff76c8f3b59.pdf.x

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota Satsuma KdV System

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called equations) , which is a generalized model of Boussinesq equation and modified Bousinesq equation , are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of equations and equations, respectively.

Yan Zhen-Ya

Papers

Financial Rogue Waves

New Jacobian Elliptic Function Solutions of Modified KdV Equation: II

New explicit and travelling wave solutions for a class of nonlinear evolution equations

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota Satsuma KdV System

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion