It has still been difficult to solve nonlinear evolution equations analytically.In this paper,we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly.Specifi...It has still been difficult to solve nonlinear evolution equations analytically.In this paper,we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly.Specifically,the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters.In particular,numerical experiments on several third-order nonlinear evolution equations,including the Korteweg-de Vries(KdV)equation,modified KdV equation,KdV-Burgers equation and Sharma-Tasso-Olver equation,demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.展开更多
Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a ...Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.展开更多
Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract ...Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.展开更多
In the present paper, we identify the integrability of the third-order nonlinear evolution equation ut = （1/2）（（uxz ＋ u）^-2）z in a Hamiltonian viewpoint. We prove that the recursion operator obtained by S.Yu. S...In the present paper, we identify the integrability of the third-order nonlinear evolution equation ut = （1/2）（（uxz ＋ u）^-2）z in a Hamiltonian viewpoint. We prove that the recursion operator obtained by S.Yu. Sakovich is hereditary, and then deduce a bi-Hamiltonian structure of the equation by using some decomposition of the hereditary operator. A hierarchy associated to the equation is also shown.展开更多
The Golstein’s strong solution formula of the second order evolution equation expands to that of the third dispersion equation by the analogy method. The semigroup expressions of its generating operator of the third ...The Golstein’s strong solution formula of the second order evolution equation expands to that of the third dispersion equation by the analogy method. The semigroup expressions of its generating operator of the third order dispersion equation are obtained, and the expression to satisfy the semigroup conditions in the three orthogonal Hilbert space of the construction is also proved. Furthermore, the necessary and sufficient conditions of the generating operator’s unitary semigroup are given .展开更多
In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operato...In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operators. We establish the existence and uniqueness ofanti-periodic solutions, which improve andgeneralize the results that have been obtained. Finally weillustrate the abstract theory by discussing a simple example of an anti-periodic problem fornonlinear partial differential equations.展开更多
Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generat...Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generated from experiments.Specifically,this framework approximates the latent solution with a deep neural network,which is trained with the constraint of underlying physical laws usually expressed by some equations.In particular,we test the effectiveness of the approach for the Burgers'equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions.The results also indicate that for soliton solutions,the model training costs significantly less time than other initial conditions.展开更多
The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn...The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn＋1＋σ2=0,where （an）, （bn}, （cn}, （qn} and （pn} are positive real sequences, β is a ratio of odd positive integers, τ1, τ2, and σ2 are positive integers. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero. Some examples are presented to illustrate the main results.展开更多
By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation wit...By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.展开更多
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evoluti...We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.展开更多
There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,...There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.展开更多
In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belon...In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.展开更多
This paper is concerned with the oscillatory properties of the third-order nonlinear delay dynamic equations of the form??on time scales , where ?is a quotient of odd positive integers. Applying the inequality techniq...This paper is concerned with the oscillatory properties of the third-order nonlinear delay dynamic equations of the form??on time scales , where ?is a quotient of odd positive integers. Applying the inequality technique we present two new sufficient conditions which ensure that every solution of equations is oscillatory or converges to zero. The results obtained improve and complement some known results in the literature.展开更多
With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixe...With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.展开更多
In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contain...In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.展开更多
In this paper,we study the weighted higher order semilinear equation in an exterior domain(-△)^(m)u=|x|^(α)g(u)in R^(N)\B_(R_(0)),where N≥1,m≥2 are integers,α>-2m,g is a continuous and nondecreasing function i...In this paper,we study the weighted higher order semilinear equation in an exterior domain(-△)^(m)u=|x|^(α)g(u)in R^(N)\B_(R_(0)),where N≥1,m≥2 are integers,α>-2m,g is a continuous and nondecreasing function in(0,+∞)and positive in(0,+∞),B_(R_(0))is the ball of the radius R0 centered at the origin.We prove that a positive supersolution of the problem which verifies(-△)_(i)u>0 in R^(N)\B_(R_(0))(i=0,…,m-1)exists if and only if N>2m and∫_(0)^(δ)g(t)/t^(2(N-m)+α/N-2m)dt<∞,,for someδ>0.We further obtain some existence and nonexistence results for the positive solution to the Dirichlet problem when g(u)=u^(p)with p>1,by using the Pohozaev identity and an embedding lemma of radial Sobolev spaces.展开更多
In view of a new idea on initial conditions, an open problem of nonlinear evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of our results is shown on an example.
This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conf...This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invaxiance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.展开更多
In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the eq...In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.展开更多
We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show tha...We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.展开更多
基金the support of the National Natural Science Foundation of China(No.11675054)the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)the Science and Technology Commission of Shanghai Municipality(No.18dz2271000)。
文摘It has still been difficult to solve nonlinear evolution equations analytically.In this paper,we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly.Specifically,the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters.In particular,numerical experiments on several third-order nonlinear evolution equations,including the Korteweg-de Vries(KdV)equation,modified KdV equation,KdV-Burgers equation and Sharma-Tasso-Olver equation,demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.
基金supported by the National Key Basic Research Project of China (973 Program)(No. 2004CB318000)
文摘Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.
基金The project supported by National Natural Science Foundation of China under Grant No. 10562002 and the Natural Science Foundation of Inner Mongolia under Grant No. 200508010103
文摘In the present paper, we identify the integrability of the third-order nonlinear evolution equation ut = （1/2）（（uxz ＋ u）^-2）z in a Hamiltonian viewpoint. We prove that the recursion operator obtained by S.Yu. Sakovich is hereditary, and then deduce a bi-Hamiltonian structure of the equation by using some decomposition of the hereditary operator. A hierarchy associated to the equation is also shown.
基金Supported by the National Nature Science Foundation of China (No. 60377021)
文摘The Golstein’s strong solution formula of the second order evolution equation expands to that of the third dispersion equation by the analogy method. The semigroup expressions of its generating operator of the third order dispersion equation are obtained, and the expression to satisfy the semigroup conditions in the three orthogonal Hilbert space of the construction is also proved. Furthermore, the necessary and sufficient conditions of the generating operator’s unitary semigroup are given .
文摘In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operators. We establish the existence and uniqueness ofanti-periodic solutions, which improve andgeneralize the results that have been obtained. Finally weillustrate the abstract theory by discussing a simple example of an anti-periodic problem fornonlinear partial differential equations.
基金supported by the National Natural Science Foundation of China(No.11675054)Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)Science and Technology Commission of Shanghai Municipality(No.18dz2271000)。
文摘Solving nonlinear evolution partial differential equations has been a longstanding computational challenge.In this paper,we present a universal paradigm of learning the system and extracting patterns from data generated from experiments.Specifically,this framework approximates the latent solution with a deep neural network,which is trained with the constraint of underlying physical laws usually expressed by some equations.In particular,we test the effectiveness of the approach for the Burgers'equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions.The results also indicate that for soliton solutions,the model training costs significantly less time than other initial conditions.
文摘The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn＋1＋σ2=0,where （an）, （bn}, （cn}, （qn} and （pn} are positive real sequences, β is a ratio of odd positive integers, τ1, τ2, and σ2 are positive integers. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero. Some examples are presented to illustrate the main results.
文摘By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.
基金We are grateful for the financial support from UKM’s research Grant GUP-2019-033。
文摘There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.
文摘In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.
文摘This paper is concerned with the oscillatory properties of the third-order nonlinear delay dynamic equations of the form??on time scales , where ?is a quotient of odd positive integers. Applying the inequality technique we present two new sufficient conditions which ensure that every solution of equations is oscillatory or converges to zero. The results obtained improve and complement some known results in the literature.
基金Supported by the National Natural Science Foundation of China(12201368,62376252)Key Project of Natural Science Foundation of Zhejiang Province(LZ22F030003)Zhejiang Province Leading Geese Plan(2024C02G1123882,2024C01SA100795).
文摘With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.
基金supported by National Natural Science Foundation of China under Grant No.10735030Natural Science Foundation of Zhejiang Province of China under Grant No.Y604056Doctoral Science Foundation of Ningbo City under Grant No.2005A61030
文摘In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.
文摘In this paper,we study the weighted higher order semilinear equation in an exterior domain(-△)^(m)u=|x|^(α)g(u)in R^(N)\B_(R_(0)),where N≥1,m≥2 are integers,α>-2m,g is a continuous and nondecreasing function in(0,+∞)and positive in(0,+∞),B_(R_(0))is the ball of the radius R0 centered at the origin.We prove that a positive supersolution of the problem which verifies(-△)_(i)u>0 in R^(N)\B_(R_(0))(i=0,…,m-1)exists if and only if N>2m and∫_(0)^(δ)g(t)/t^(2(N-m)+α/N-2m)dt<∞,,for someδ>0.We further obtain some existence and nonexistence results for the positive solution to the Dirichlet problem when g(u)=u^(p)with p>1,by using the Pohozaev identity and an embedding lemma of radial Sobolev spaces.
基金supported by TWAS,UNESO and AMSS in Chinese AcademyThe research of the third author is partially supported by NSFC(11001239)
文摘In view of a new idea on initial conditions, an open problem of nonlinear evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of our results is shown on an example.
基金Project supported by the Graduate Students Innovative Foundation of China University of Petroleum (East China) (Grant NoS2009-19)
文摘This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invaxiance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.
文摘In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.
基金Supported by the National Natural Science Foundation of China(61473340)Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province(F703108L02)。
文摘We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.
