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Abstract: In this paper,we show that for any given planar cubic algebraic curves defined by a quadratic Hamiltonian vector field,we can always have their exact explicit parametric representations.We use a model of microstructured solid to show an application of our conclusions.
Key words: cubic algebraic curve; planar quadratic Hamiltonian system; exact solution; traveling wave; model of microstructured solid
CLC number: O 241.8, O 152.5 Document code: A
Article ID: 10005137(2014)05045608
2010 Mathematics Subject Classification: 14H50, 34C25, 35Q72
Received date: 20140330
Foundation item: Supported by the National Natural Science Foundation of China(11471289,11162020)
Corresponding author: Jibin Li,Professor,Email:lijb@zjnu.cn; jibinli@gmail.com1 Introduction
We are interested in the study of planar vector fields because they occur very often in applications.Indeed,such equations appear in modelling chemical reactions,population dynamics,travelling wave systems of nonlinear evolution equations in mathematical physics and in many other areas of applied mathematics and mechanics.
It is known that a planar quadratic Hamiltonian vector fielddxdt=Fy, dydt=-Fx
(1)defines a family of cubic algebraic (2)For every given real number h,the curves defined by F(x,y)=0 are called the level curves or the orbits of system (1).
Can we have the exact explicit parametric representation x=x(t),y=y(t) for a given cubic curve? This is an interesting problem.In fact,for a given nonlinear wave equation,if its corresponding traveling wave system is a planar quadratic integrable system,to obtain exact traveling wave solutions,we need to solve the above problem.
In this paper,we show that for any given planar cubic algebraic curves,we can always have their exact explicit parametric representations.To obtain this conclusion,we first introduce the classification of cubic curves given by Newton and a method that makes a general cubic curves become corresponding classification types.Second,for the (A)-type curves,we show that it is easy to get the exact explicit parametric representations.In section 3,we use the normal form given by Horozov & Iliev［1］ to prove that any cubic curves defined by a quadratic Hamiltonian system can get their exact explicit parametric representations.As an example,we discuss the traveling wave solutions for a model of microstructured solid.We give the formulas,for which we can obtain the exact explicit parametric representations of the traveling wave solutions. 2 Preliminary:cubic algebraic curves
In 1704,Newton classified the cubic algebraic curves of the form(2) into the following four types［2-3］:(A) Cubic curves with more branches(at most three):xy2+ey=ax3+bx2+cx+d.
(B) Parabolichyperbolic curves(two branches):xy=ax3+bx2+cx+d.
(C) Divergent parabola(one or two branches):y2=ax3+bx2+cx+d.
(D) Cubic parabola(one branch):y=ax3+bx2+cx+d. Question 1 How to make form(2) become one of the above four types?
We write that f3(x,y)=Ax3+3Bx2y+3Cxy2+Dy3, f2(x,y)=3Ex2+6Fxy+3Gy2 , f1(x,y)=Hx+Iy.
4 An example of application:exact travelling wave solutions for the model of microstructured solids
Microstructured materials like alloys,crystallites,ceramics,functionally graded materials,etc.,have gained wide applications.There are a lot of modelling of wave propagation in such materials［4-6］.In［7］,the author presented a mathematical model for longitudinal waves in the 1D setting which describes nonlinear wave motion in microstructured solids.The governing system is the following:ρ0utt=auxx+Nuxuxx+AΨx, IΨtt=CΨxx+MΨxΨxx-Aux-BΨ,
(14)where u denotes the longitudinal(macro)displacement and Ψ is the microdeformation (according to the Mindlin mode) or the internal variable (according to the concept of internal variables).Further,ρ0 is the density and I inertia of the microstructure,while,a,A,B,C,N,M are the material parameters specifying the free energy function.After introducing the dimensionless variables U,X,T and applying the “slaving principle”,system(14) is reduced to one equation［8］:UTT=1-cA2c02UXX+12kN(UX2)X+cA2cB2UTT-c21c20UXXXX+12KM(UXX2)XX,
(15) where c0,c1,cA,cB are velocities and kN,kM are the parameters expressing the strengths of physical nonlinearities on macroand microscale,respectively.
References:
［1］ E. Horozov,D. Iliev.On the number of limit cycles in perturbations of quadratic Hamiltonian systems［J］.Proc. London Math. Soc.,1994,69(3):198-224.
［2］ S. B. Mulay.Classification of plane cubic curves,Advances in commutative ring theory(fez,1997),461-482,Lecture Notice in Pure and Appl. Math.,Dekker,New York,1999,205.
［3］ A. S. Smogorzhevskii,E. S. Stolova,Handbook on the theory of planar curves of the third order［M］.Moscow:Fizmatgiz,1961.
［4］ A. C. Eringgen.Microcontinuum Field Theories.I Foundations and Solids［M］.New York:Springer,1999.
［5］ R. Phillips.Crystal,Defects and Microstructures.Modelling Across Scales,Cambridge University Press,Combridge,2001.
［6］ S. Suresh,A. Mortensen.Fundamentals of Functionally Graded Materials［M］.London:IOM Comm. Ltd.
［7］ J. Engelbrecht.Nonlinear wave motion and complexity［J］.Proceedings of the Estonian Academy of Sciences,2010,59(2):66-71.
［8］ J. Janno,J. Engelbrecht.Solitary waves in nonlinear microstructured materials［J］.J. Phys. A.:Math Gen,2005,38:5159-5172.
［9］ J. B. Li,G. R. Chen.Exact travelling wave solutions and their bifurcations for the model of microstructured solids［J］.Int. J. Bifurcation and Chaos,2013,23(1),1350009.
(Zhenzhen Feng)
Key words: cubic algebraic curve; planar quadratic Hamiltonian system; exact solution; traveling wave; model of microstructured solid
CLC number: O 241.8, O 152.5 Document code: A
Article ID: 10005137(2014)05045608
2010 Mathematics Subject Classification: 14H50, 34C25, 35Q72
Received date: 20140330
Foundation item: Supported by the National Natural Science Foundation of China(11471289,11162020)
Corresponding author: Jibin Li,Professor,Email:lijb@zjnu.cn; jibinli@gmail.com1 Introduction
We are interested in the study of planar vector fields because they occur very often in applications.Indeed,such equations appear in modelling chemical reactions,population dynamics,travelling wave systems of nonlinear evolution equations in mathematical physics and in many other areas of applied mathematics and mechanics.
It is known that a planar quadratic Hamiltonian vector fielddxdt=Fy, dydt=-Fx
(1)defines a family of cubic algebraic (2)For every given real number h,the curves defined by F(x,y)=0 are called the level curves or the orbits of system (1).
Can we have the exact explicit parametric representation x=x(t),y=y(t) for a given cubic curve? This is an interesting problem.In fact,for a given nonlinear wave equation,if its corresponding traveling wave system is a planar quadratic integrable system,to obtain exact traveling wave solutions,we need to solve the above problem.
In this paper,we show that for any given planar cubic algebraic curves,we can always have their exact explicit parametric representations.To obtain this conclusion,we first introduce the classification of cubic curves given by Newton and a method that makes a general cubic curves become corresponding classification types.Second,for the (A)-type curves,we show that it is easy to get the exact explicit parametric representations.In section 3,we use the normal form given by Horozov & Iliev［1］ to prove that any cubic curves defined by a quadratic Hamiltonian system can get their exact explicit parametric representations.As an example,we discuss the traveling wave solutions for a model of microstructured solid.We give the formulas,for which we can obtain the exact explicit parametric representations of the traveling wave solutions. 2 Preliminary:cubic algebraic curves
In 1704,Newton classified the cubic algebraic curves of the form(2) into the following four types［2-3］:(A) Cubic curves with more branches(at most three):xy2+ey=ax3+bx2+cx+d.
(B) Parabolichyperbolic curves(two branches):xy=ax3+bx2+cx+d.
(C) Divergent parabola(one or two branches):y2=ax3+bx2+cx+d.
(D) Cubic parabola(one branch):y=ax3+bx2+cx+d. Question 1 How to make form(2) become one of the above four types?
We write that f3(x,y)=Ax3+3Bx2y+3Cxy2+Dy3, f2(x,y)=3Ex2+6Fxy+3Gy2 , f1(x,y)=Hx+Iy.
4 An example of application:exact travelling wave solutions for the model of microstructured solids
Microstructured materials like alloys,crystallites,ceramics,functionally graded materials,etc.,have gained wide applications.There are a lot of modelling of wave propagation in such materials［4-6］.In［7］,the author presented a mathematical model for longitudinal waves in the 1D setting which describes nonlinear wave motion in microstructured solids.The governing system is the following:ρ0utt=auxx+Nuxuxx+AΨx, IΨtt=CΨxx+MΨxΨxx-Aux-BΨ,
(14)where u denotes the longitudinal(macro)displacement and Ψ is the microdeformation (according to the Mindlin mode) or the internal variable (according to the concept of internal variables).Further,ρ0 is the density and I inertia of the microstructure,while,a,A,B,C,N,M are the material parameters specifying the free energy function.After introducing the dimensionless variables U,X,T and applying the “slaving principle”,system(14) is reduced to one equation［8］:UTT=1-cA2c02UXX+12kN(UX2)X+cA2cB2UTT-c21c20UXXXX+12KM(UXX2)XX,
(15) where c0,c1,cA,cB are velocities and kN,kM are the parameters expressing the strengths of physical nonlinearities on macroand microscale,respectively.
References:
［1］ E. Horozov,D. Iliev.On the number of limit cycles in perturbations of quadratic Hamiltonian systems［J］.Proc. London Math. Soc.,1994,69(3):198-224.
［2］ S. B. Mulay.Classification of plane cubic curves,Advances in commutative ring theory(fez,1997),461-482,Lecture Notice in Pure and Appl. Math.,Dekker,New York,1999,205.
［3］ A. S. Smogorzhevskii,E. S. Stolova,Handbook on the theory of planar curves of the third order［M］.Moscow:Fizmatgiz,1961.
［4］ A. C. Eringgen.Microcontinuum Field Theories.I Foundations and Solids［M］.New York:Springer,1999.
［5］ R. Phillips.Crystal,Defects and Microstructures.Modelling Across Scales,Cambridge University Press,Combridge,2001.
［6］ S. Suresh,A. Mortensen.Fundamentals of Functionally Graded Materials［M］.London:IOM Comm. Ltd.
［7］ J. Engelbrecht.Nonlinear wave motion and complexity［J］.Proceedings of the Estonian Academy of Sciences,2010,59(2):66-71.
［8］ J. Janno,J. Engelbrecht.Solitary waves in nonlinear microstructured materials［J］.J. Phys. A.:Math Gen,2005,38:5159-5172.
［9］ J. B. Li,G. R. Chen.Exact travelling wave solutions and their bifurcations for the model of microstructured solids［J］.Int. J. Bifurcation and Chaos,2013,23(1),1350009.
(Zhenzhen Feng)