In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order, They first give the well posedness of general discontinuous boundary...In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order, They first give the well posedness of general discontinuous boundary value problems, reduce the discontinuous boundary value problems to a variation problem, and then find the numerical solutions of above problem by the finite element method. Finally authors give some error-estimates of the foregoing numerical solutions.展开更多
In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-nes...In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.展开更多
This paper considers the Riemann-Hilbert problem for linear mixed(elliptichyperbolic) complex equations of first order with degenerate curve in a simply connected domain. We first give the representation theorem and...This paper considers the Riemann-Hilbert problem for linear mixed(elliptichyperbolic) complex equations of first order with degenerate curve in a simply connected domain. We first give the representation theorem and uniqueness of solutions for such boundary value problem. Then by using the methods of successive iteration and parameter extension, the existence of solutions for this problem is proved.展开更多
In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.Th...In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem(0.1) and(0.2).展开更多
Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained thro...Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.展开更多
We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem ...We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.展开更多
This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed (elliptic- hyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem an...This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed (elliptic- hyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of HSlder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.展开更多
In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its mo...In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.展开更多
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj(z)f1(λj1)(z+cj)=R2(z,f2(z)),j=1∑nβj(z)f2(λj2)(z+cj)=R1(z,f1(z)). where λij (j = 1, 2,…, n; i = 1, 2) are finite non-negative integers, and cj (j = 1, 2,… , n) are distinct, nonzero complex numbers, αj(z), βj(z) (j = 1,2,… ,n) are small functions relative to fi(z) (i =1, 2) respectively, Ri(z, f(z)) (i = 1, 2) are rational in fi(z) (i =1, 2) with coefficients which are small functions of fi(z) (i = 1, 2) respectively.展开更多
In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter ...In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter and Laine's result concerning complex differential equations to the systems of algebraic differential equations.展开更多
The wave propagation in the one-dimensional complex Ginzbur-Landau equation (CGLE) is studied by considering a wave source at the system boundary. A special propagation region, which is an island-shaped zone surroun...The wave propagation in the one-dimensional complex Ginzbur-Landau equation (CGLE) is studied by considering a wave source at the system boundary. A special propagation region, which is an island-shaped zone surrounded by the defect turbulence in the system parameter space, is observed in our numerical experiment. The wave signal spreads in the whole space with a novel amplitude wave pattern in the area. The relevant factors of the pattern formation, such as the wave speed, the maximum propagating distance and the oscillatory frequency, are studied in detail. The stability and the generality of the region are testified by adopting various initial conditions. This finding of the amplitude pattern extends the wave propagation region in the parameter space and presents a new signal transmission mode, and is therefore expected to be of much importance.展开更多
By using the three-dimensional complex Ginzburg--Landau equation with cubic--quintic nonlinearity, this paper numerically investigates the interactions between optical bullets with different velocities in a dissipativ...By using the three-dimensional complex Ginzburg--Landau equation with cubic--quintic nonlinearity, this paper numerically investigates the interactions between optical bullets with different velocities in a dissipative system. The results reveal an abundance of interesting behaviours relating to the velocities of bullets: merging of the optical bullets into a single one at small velocities; periodic collisions at large velocities and disappearance of two bullets after several collisions in an intermediate region of velocity. Finally, it also reports that an extra bullet derives from the collision of optical bullets when optical bullets are at small velocities but with high energies.展开更多
For entire or meromorphic function f,a value θ∈[0,2π)is called a Julia limiting direction if there is an unbounded sequence{z_(n)}in the Julia set satisfying limn→∞ arg z_(n)=θ.Our main result is on the entire s...For entire or meromorphic function f,a value θ∈[0,2π)is called a Julia limiting direction if there is an unbounded sequence{z_(n)}in the Julia set satisfying limn→∞ arg z_(n)=θ.Our main result is on the entire solution f of P(z,f)+F(z)f^(s)=0,where P(z,f)is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F,with the integer s being no more than the minimum degree of all differential monomials in P(z,f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.展开更多
In this paper, exact solutions are derived for four coupled complex nonlinear different equations by using simplified transformation method and algebraic equations.
The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theore...The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.展开更多
The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to t...The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.展开更多
So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. Bu...So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.展开更多
We prove the existence of a uniform initial datum whose solution decays, in var- ious Lp spaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of vari...We prove the existence of a uniform initial datum whose solution decays, in var- ious Lp spaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.展开更多
The purpose of this work is to find new soliton solutions of the complex Ginzburg–Landau equation(GLE)with Kerr law non-linearity.The considered equation is an imperative nonlinear partial differential equation(PDE)i...The purpose of this work is to find new soliton solutions of the complex Ginzburg–Landau equation(GLE)with Kerr law non-linearity.The considered equation is an imperative nonlinear partial differential equation(PDE)in the field of physics.The applications of complex GLE can be found in optics,plasma and other related fields.The modified extended tanh technique with Riccati equation is applied to solve the Complex GLE.The results are presented under a suitable choice for the values of parameters.Figures are shown using the three and two-dimensional plots to represent the shape of the solution in real,and imaginary parts in order to discuss the similarities and difference between them.The graphical representation of the results depicts the typical behavior of soliton solutions.The obtained soliton solutions are of different forms,such as,hyperbolic and trigonometric functions.The results presented in this paper are novel and reported first time in the literature.Simulation results establish the validity and applicability of the suggested technique for the complex GLE.The suggested method with symbolic computational software such as,Mathematica and Maple,is proven as an effective way to acquire the soliton solutions of nonlinear partial differential equations(PDEs)as well as complex PDEs.展开更多
文摘In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order, They first give the well posedness of general discontinuous boundary value problems, reduce the discontinuous boundary value problems to a variation problem, and then find the numerical solutions of above problem by the finite element method. Finally authors give some error-estimates of the foregoing numerical solutions.
文摘In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.
基金Supported by the National Natural Science Foundation of China (10971224)
文摘This paper considers the Riemann-Hilbert problem for linear mixed(elliptichyperbolic) complex equations of first order with degenerate curve in a simply connected domain. We first give the representation theorem and uniqueness of solutions for such boundary value problem. Then by using the methods of successive iteration and parameter extension, the existence of solutions for this problem is proved.
文摘In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem(0.1) and(0.2).
基金Project supported by the National Natural Science Foundation of China (Grant No. 12161061)the Fundamental Research Funds for the Inner Mongolia University of Finance and Economics (Grant No. NCYWT23036)+2 种基金the Young Innovative and Entrepreneurial Talents of the Inner Mongolia Grassland Talents Project in 2022,Autonomous Region “Five Major Tasks” Research Special Project for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. NCXWD2422)High Quality Research Achievement Cultivation Fund for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. GZCG2426)the Talent Development Fund of Inner Mongolia Autonomous Region, China。
文摘Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.
基金Project supported by the National Natural Science Foundation of China(Grant No.11975145)the Program for Science&Technology Innovation Talents in Universities of Henan Province,China(Grant No.22HASTIT019)+2 种基金the Natural Science Foundation of Henan,China(Grant No.202300410524)the Science and Technique Project of Henan,China(Grant No.212102310397)the Academic Degrees&Graduate Education Reform Project of Henan Province,China(Grant No.2021SJGLX219Y)。
文摘We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11021161 and 10928102973 Program of China under Grant No.2011CB80800+2 种基金Chinese Academy of Sciences under Grant No.kjcx-yw-s7project grant of "Center for Research and Applications in Plasma Physics and Pulsed Power Technology,PBCT-Chile-ACT 26"Direcci'on de Programas de Investigaci'on,Universidad de Talca,Chile
文摘This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed (elliptic- hyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of HSlder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.
文摘In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj(z)f1(λj1)(z+cj)=R2(z,f2(z)),j=1∑nβj(z)f2(λj2)(z+cj)=R1(z,f1(z)). where λij (j = 1, 2,…, n; i = 1, 2) are finite non-negative integers, and cj (j = 1, 2,… , n) are distinct, nonzero complex numbers, αj(z), βj(z) (j = 1,2,… ,n) are small functions relative to fi(z) (i =1, 2) respectively, Ri(z, f(z)) (i = 1, 2) are rational in fi(z) (i =1, 2) with coefficients which are small functions of fi(z) (i = 1, 2) respectively.
基金Project Supported by the Natural Science Foundation of China(10471065)the Natural Science Foundation of Guangdong Province(04010474)
文摘In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter and Laine's result concerning complex differential equations to the systems of algebraic differential equations.
文摘The wave propagation in the one-dimensional complex Ginzbur-Landau equation (CGLE) is studied by considering a wave source at the system boundary. A special propagation region, which is an island-shaped zone surrounded by the defect turbulence in the system parameter space, is observed in our numerical experiment. The wave signal spreads in the whole space with a novel amplitude wave pattern in the area. The relevant factors of the pattern formation, such as the wave speed, the maximum propagating distance and the oscillatory frequency, are studied in detail. The stability and the generality of the region are testified by adopting various initial conditions. This finding of the amplitude pattern extends the wave propagation region in the parameter space and presents a new signal transmission mode, and is therefore expected to be of much importance.
基金Project supported by the Key Project of the Educational Department of Hunan Province of China (Grant No. 04A058)the General Project of the Educational Department of Hunan Province of China (Grant No. 07C754)
文摘By using the three-dimensional complex Ginzburg--Landau equation with cubic--quintic nonlinearity, this paper numerically investigates the interactions between optical bullets with different velocities in a dissipative system. The results reveal an abundance of interesting behaviours relating to the velocities of bullets: merging of the optical bullets into a single one at small velocities; periodic collisions at large velocities and disappearance of two bullets after several collisions in an intermediate region of velocity. Finally, it also reports that an extra bullet derives from the collision of optical bullets when optical bullets are at small velocities but with high energies.
基金This work was supported by the National Natural Science Foundation of China(11771090,11901311)Natural Sciences Foundation of Shanghai(17ZR1402900).
文摘For entire or meromorphic function f,a value θ∈[0,2π)is called a Julia limiting direction if there is an unbounded sequence{z_(n)}in the Julia set satisfying limn→∞ arg z_(n)=θ.Our main result is on the entire solution f of P(z,f)+F(z)f^(s)=0,where P(z,f)is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F,with the integer s being no more than the minimum degree of all differential monomials in P(z,f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.
文摘In this paper, exact solutions are derived for four coupled complex nonlinear different equations by using simplified transformation method and algebraic equations.
基金Supported by Guangdong Natural Science Foundation(2015A030313628,S2012010010376)Training plan for Distinguished Young Teachers in Higher Education of Guangdong(Yqgdufe1405)+1 种基金Guangdong Education Science Planning Project(2014GXJK091,GDJG20142304)the National Natural Science Foundation of China(11301140,11101096)
文摘The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.
文摘The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
基金Project supported by the National Natural Science Foundation of China (Grant No 10274003) and the Department of Science and Technology of China.Acknowledgement We thank Cheng X, Wang C and Wang S for helpful discussion.
文摘The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10575087 and 10875106)
文摘So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.
基金Supported by NSFC(11271322,11271105)ZJNSF(LQ14A010011)
文摘We prove the existence of a uniform initial datum whose solution decays, in var- ious Lp spaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.
基金the National Natural Science Foundation of China(Grant Nos.11971142,11871202,61673169,11701176,11626101,11601485).YMC received the grant for this work.
文摘The purpose of this work is to find new soliton solutions of the complex Ginzburg–Landau equation(GLE)with Kerr law non-linearity.The considered equation is an imperative nonlinear partial differential equation(PDE)in the field of physics.The applications of complex GLE can be found in optics,plasma and other related fields.The modified extended tanh technique with Riccati equation is applied to solve the Complex GLE.The results are presented under a suitable choice for the values of parameters.Figures are shown using the three and two-dimensional plots to represent the shape of the solution in real,and imaginary parts in order to discuss the similarities and difference between them.The graphical representation of the results depicts the typical behavior of soliton solutions.The obtained soliton solutions are of different forms,such as,hyperbolic and trigonometric functions.The results presented in this paper are novel and reported first time in the literature.Simulation results establish the validity and applicability of the suggested technique for the complex GLE.The suggested method with symbolic computational software such as,Mathematica and Maple,is proven as an effective way to acquire the soliton solutions of nonlinear partial differential equations(PDEs)as well as complex PDEs.