SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

Validation of a Questionnaire to Diagnose the Baumann Skin Type in All Ethnicities and in Various Geographic Locations

The Baumann Skin Typing System diagnoses patients as having one of 16 skin types based on their answers to a validated questionnaire [i] known as the Baumann Skin Type Indicator [ii]. The BSTI questionnaire has been tested over the last decade on over 200,000 people of various ages and ethnicities in different geographic locations around the world. In this study, data were collected from 52,862 patients to compare skin type prevalence between those who presented to doctor’s offices and those who took the quiz without supervision online. The most common skin types varied only slightly between patients that took the quiz online and those that completed the questionnaire in their doctor’s office. This indicates that the prevalence of skin types seen in the doctor’s office is similar to that in the general population and that supervision is not necessary to get an accurate result on the BSTI. [iii] In addition, comparison of data gathered in China, Korea, and the US did not show a significant difference in skin type prevalence between Asian and Caucasian skin types. [iv] This study demonstrates that the English version of the BSTI is valid for English speaking patients online, and in doctors’ offices in the US, China and Korea.

Integrability Tests and Some New Soliton Solutions of an Extended Potential Boiti-Leon-Manna-Pempinelli Equation

This paper is devoted to the study of a (2 + 1)-dimensional extended Potential Boiti-Leon-Manna-Pempinelli equation. Firstly, By means of the standard Weiss Tabor Carnevale approach and Kruskal’s simplification, we prove the painlevé non integrability of the equation. Secondly, A new breather solution and lump type solution are obtained based on the parameter limit method and Hirota’s bilinear method. Besides, some interaction behavior between lump type solution and N-soliton solutions (N is any positive integer) are studied. We construct the existence theorem of the interaction solution and give the process of calculation and proof. We also give a concrete example to illustrate the effectiveness of the theorem, and some spatial structure figures are displayed to reflect the evolutionary behavior of the interaction solutions with the change of soliton number N and time t.

Impulse Spatial-Temporal Domains in Semiconductor Laser with Feedback

An initial value boundary problem for system of diffusion equations with delay arguments and dynamic nonlinear boundary conditions is considered. The problem describes evolution of the carrier density and the radiation density in the semiconductor laser or laser diodes with “memory” and with feedback. It is shown that the boundary problem can be reduced to a system of difference equations with continuous time. For large times, solutions of these equations tend to piecewise constant asymptotic periodic wave functions which represent chain of shock waves with finite or infinite points of discontinuities on a period. Applications to the optical systems with linear media and nonlinear surface optical properties with feedback have been done. The results are compared with the experiment.

Source Type Solutions of a Fourth Order Degenerate Parabolic Equation

In this paper, we study a generalized thin film equation which is relevant to capillary driven flows of thin films of power-law fluids. We prove that the generalized thin film equation in dimension d ≥ 2 has a unique C^1 source type radial self-similar nonnegative solution if 0 〈 n 〈 2p - 1 and has no solution of this type if n ≥ 2p - 1.

Existence of ground state solutions of Nehari-Pankov type to Schr?dinger systems

This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE 0F PERIODIC SOLUTIONS OF A TYPE OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.

Multiply Kink and Anti-Kink Solutions for a Coupled Camassa–Holm Type Equation

This article amis at revealing dynamical behavior of a coupled Camassa–Holm type equation, which was proposed by Geng and Wang based on a 4×4 matrix spectral problem with two potentials. Its kink and anti-kink solutions are presented explicitly. In particular, some exact multi-kink and anti-kink wave solutions are discussed and under some conditions, the kink and anti-kinks look like hat-shape solitons. The dynamic characters of the obtained solutions are investigated by figures. The method used in this paper can be widely applied to looking for the multi-kinks for Camassa–Holm type equations possessing cubic nonlinearity.

MODELLING AND NUMERICAL SOLUTIONS OF A GAUGE PERIODIC TIME DEPENDENT GINZBURG-LANDAU MODEL FOR TYPE-Ⅱ SUPERCONDUCTORS

In this paper we seek the solutions of the time dependent Ginzburg-Landau model for type-Ⅱ superconductors such that the associated physical observables are spatially periodic with respect to some lattice whose basic lattice cell is not necessarily rectangular. After appropriately foring the gange, the model can be formulated as a system of nonlinear parabolic partial differential equations with quasi-periodic boundary conditions. We first give some results concerning the existence, uniqueness and regularity of solutions and then we propose a semiimplicit finite element scheme solving the system of nonlinear partial dmerential equations and show the optimal error estimates both in the L2 and energy norm.We also report on some numerical results at the end of the paper.

TYPES AND CRITERIA OF NONOSCILLATORY SOLUTIONS FOR A CLASS OF SECOND ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONPOSITIVE COEFFICIENTS

In this paper we discuss the types and criteria of nonoscillatory solutions for the fol-lowing second order neutral functional differential equation with nonpositive coefficients

Non-Nehari manifold method for asymptotically periodic Schrodinger equations

We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.

QUALITATIVE ANALYSIS FOR A THIRD-ORDER DIFFERENTIAL EQUATION IN A MODEL OF CHEMICAL SYSTEMS

In this paper a mathematical model of chemical systems is investigated.We present the conditions for the existence and local stability of the steady statesand the periodic solution of the Hopf type.Specifically,we show by using an ana-lytical method that there may exist two or four Hopf bifurcation points separatedat a finite distance from each other;at the same time,a technique for studying theHopf bifurcation value is given.

Exact Solitary Wave Solutions of Nonlinear Evolution Equations with a Positive Fractional Power Term

The bounded and smooth solitary wave solutions of 10 nonlinear evolution equations with a positive fractional power term of dependent variable are successfully obtained by homogeneous balance principle and with the aid of sub-ODEs that admits a solution of sech-power or tanh-power type.In the special cases that the fractional power equals to 1 and 2,the solitary wave solutions of more than 10 important model equations arisen from mathematical physics are easily rediscovered.

Non-Nehari Manifold Method for Periodic Discrete Superlinear Schr(o|¨)dinger Equation

We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.

HOPF BIFURCATION AND ANALYSIS OF EQUILIBRIUM FOR A THIRD-ORDER DIFFERENTIAL EQUATION IN A MODEL OF COMPETITION

In this paper, a mathematical model of competition between plasmid-bearing and plasmidfree organisms in a chemostat with an inhibitor is investigated. The model is in the form of a system of nonlinear differential equations. By using qualitative methods, the conditions for the existence and local stability of the equilibria are obtained. The existence and stability of periodic solutions of the Hopf type are studied. Numerical simulations about the Hop f bifurcation value and Hopf limit cycle are also given.