The equation (1) arises from an intrinsic (arc-length preserving) invariant planar curve flow in Euclidean geometry and it can be regarded as a Euclidean-invariant version of the Camassa-Holm equation in [1]. It has the form of a modified Camassa-Holm equation with cubic nonlinearity. By Fuchssteiner [2] and Olver and Rosenau[3], it can be derived as a new integrable system by applying the gene...

We derive the evolution equations of parton distribution functions appropriate in different kinematic regions in a unified and simple way using the resummation technique. They include the Gribov-Lipatov-Altarelli-Parisi equation for large momentum transfer Q, the Balitskii-Fadin-Kuraev-Lipatov equation for a small Bjorken variable x, and the Ciafaloni-Catani-FioraniMarchesini equation which uni...

Trajectory-based approaches to quantum mechanics include the de Broglie-Bohm interpretation and Nelson’s stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schrödinger equation into a continuity equation and a modified Hamilton-Jacobi equation, fails for some quantum states. A very simple example is provid...

In this paper the interaction of attosecond laser pulses with matter is investigated. The scattering and potential motion of heat carriers as well as the external force are considered. Depending on the ratio of the scatterings and potential motion the heat transport is described by the thermal forced Klein-Gordon or thermal modified telegraph equation. For thermal Heisenberg type relation V τ ∼...

In this paper, using modified FF-expansion method, we present some explicit formulas of exact traveling wave solutions for the foam drainage equation. A modified FF-expansion method is proposed by taking full advantages of FF-expansion method and Riccati equation in seeking exact solutions of non-linear partial differential equations.

This paper is concerned with the Cauchy problem of the modified Kawahara equation. By using the Fourier restriction norm method introduced by Bourgain, and using the I-method as well as the L 2 conservation law, we prove that the modified Kawahara equation is globally well-posed for the initial data in the Sobolev space H s (R) with s > − 3 22 .

For analyzing the harvest of the Gulf of Mexico shrimp fishery, Griffin et al. (1976) have developed an equation that relates shrimp yield to freshwater discharge of the Mississippi River and fishing effort of Gulf shrimp vessels. The yield equation (referred to as Griffin's equation) is a modified Spillman production function (Heady and Dillon 1972). The Spillman function had its origin in agr...

Trajectory-based approaches to quantum mechanics include the de Broglie-Bohm interpretation and Nelson’s stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schrödinger equation into a continuity equation and a modified Hamilton-Jacobi equation, fails for some quantum states. A very simple example is provid...

In this paper we study the half-linear differential equation ( r(t)Φp(x ′) ) ′ + c(t)Φp(x) = 0, where Φp(x) = |x|p−2x, p > 1. Using modified Riccati technique and suitable local estimates for terms in modified Riccati equation we derive new characterization of principal solution and new nonoscillation criteria.

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation m...