Non-integrability Of Some Physical And Chemical Models

Differential equations is an important branch of modern mathematics,which is an effective tool for people to solve practical problems,such as Geometrical,Mechanics,Physics,Electronic Technology,Automatic control,Aerospace,Biology,Economics,etc.Quantitative study variation law of practical problems,properly simplify and hypothesize the problem,establish a mathematical model,when the problem involves the variable rate of variable,the model is a differential equation.Such as radium decay,thermal conduction,spring vibration,free fall and other issues.Normally,the differential equations can describe almost the problems of the rate of variable.Integrability is a fundamental problem in the field of differential equations research,which main concern to find enough equations for the first integral or conservation of the law,so that it can be expressed in the form of the general solution.Over the years,many mathematicians,including the famous mathematician Poincar?e,built and developed many theories and methods of studying integrable,such as,Noether symmetry,Darboux integral,Lie symmetry,Painlev'e analysis,Lax pair,Carlemann embedding method,Darboux integrability theory and quasi-homogeneous system theory,and get a series of important results.In this paper,we focuses on the integrability in the dynamic model of chemistry and physics.The paper structure is as follows:The first chapter will briefly introduce the integrable and research theory and methods;The second chapter mainly introduces two methods of study the integrable equations of differential equations: semi-quasi-homogeneous system theory and Poincarétheory of nonintegrable and generalized correlation,and related results;In the third chapter,we study the integrability of a class of chemical reaction kinetic differential equations?the Einstein-Yang-Mills equation and the Szekeres system.The main results are as follows:1.Consider the following kinetic differential equation which describe the chemical reaction where x1,x2,x3,x4,x5 denote the concentration of reactant A,B,C,D,E.We get the following result.Theorem 0.1 If there exist m,n ? C such that then the number of functionally independent analytic first integrals of system(1)is less than or equal to three,whereA =-(c1m2+ c3 n + c2+ c4),B =c2m4-2c3c1m2 n + 2c1(c2-c4)m2+ c23n2+ 2c3(c2-c4)n + c2(c2-2c4+ 1).Theorem 0.2 If there exist m,n ? C,such that then the number of functionally independent analytic first integrals of system(1)is less than or equal to three.2.For Einstein-Yang-Mills equation we obtained the following result.Theorem 0.3 If there exist m,n ? C such that then the number of functionally independent analytic first integrals of system(2)is less than or equal to four,where.3.For Szekeres system we getTheorem 0.4 The absence of analytic first integral in system Szekeres.