In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic s...In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic solution was obtained.展开更多
A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimen...A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.展开更多
Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integra...Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integrable systems in higher dimensions.Recent studies have shown that abundant higher-dimensional integrable systems can be constructed from(1+1)-dimensional integrable systems by using a deformation algorithm.Here we establish a new(2+1)-dimensional Chen-Lee-Liu(C-L-L)equation using the deformation algorithm from the(1+1)-dimensional C-L-L equation.The new system is integrable with its Lax pair obtained by applying the deformation algorithm to that of the(1+1)-dimension.It is challenging to obtain the exact solutions for the new integrable system because the new system combines both the original C-L-L equation and its reciprocal transformation.The traveling wave solutions are derived in implicit function expression,and some asymmetry peakon solutions are found.展开更多
Abstract In this paper the implicit obstacle problem of fully nonlinear second order elliptic equations associated with impulsive control problem are investigated.The comparion principle for viscosity solutions is pro...Abstract In this paper the implicit obstacle problem of fully nonlinear second order elliptic equations associated with impulsive control problem are investigated.The comparion principle for viscosity solutions is proved,the existence and uniqueness results are disscussed.展开更多
In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem in the presence of an upper solution βand lower...In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem in the presence of an upper solution βand lower solution a with β a.展开更多
An analysis of logistic stochastic differential equations(SDEs)with general power-law and driven by a Wiener process is conducted.We prove existence of unique,strong Markovian,continuous solutions.The solutions live(a...An analysis of logistic stochastic differential equations(SDEs)with general power-law and driven by a Wiener process is conducted.We prove existence of unique,strong Markovian,continuous solutions.The solutions live(a.s.)on bounded domains D=[0,K]required by applications to biology,ecology and physics with nonrandom threshold parameter K>0(i.e.the maximum carrying constant).Moreover,we present and justify nonstandard numerical methods constructed by specified balanced implicit methods(BIMs).Their weak and L^(p)-convergence follows from the fact that these methods with local Lipschitz-continuous coefficients of logistic SDEs“produce”positive numerical approximations on bounded domain[0,K](a.s.).As commonly known,standard numerical methods such as Taylor-type ones for SDEs fail to do that.Finally,asymptotic stability of nontrivial equilibria x_(∗)=K is proven for both continuous time logistic SDEs and discrete time approximations by BIMs.We exploit the technique of positive,sufficiently smooth and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.展开更多
文摘In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic solution was obtained.
基金support of the National Natural Science Foundation of China(Nos.12275144,12235007 and 11975131)the K C Wong Magna Fund at Ningbo University。
文摘A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12275144,12235007,and 11975131)K.C.Wong Magna Fund in Ningbo University。
文摘Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integrable systems in higher dimensions.Recent studies have shown that abundant higher-dimensional integrable systems can be constructed from(1+1)-dimensional integrable systems by using a deformation algorithm.Here we establish a new(2+1)-dimensional Chen-Lee-Liu(C-L-L)equation using the deformation algorithm from the(1+1)-dimensional C-L-L equation.The new system is integrable with its Lax pair obtained by applying the deformation algorithm to that of the(1+1)-dimension.It is challenging to obtain the exact solutions for the new integrable system because the new system combines both the original C-L-L equation and its reciprocal transformation.The traveling wave solutions are derived in implicit function expression,and some asymmetry peakon solutions are found.
文摘Abstract In this paper the implicit obstacle problem of fully nonlinear second order elliptic equations associated with impulsive control problem are investigated.The comparion principle for viscosity solutions is proved,the existence and uniqueness results are disscussed.
文摘In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem in the presence of an upper solution βand lower solution a with β a.
文摘An analysis of logistic stochastic differential equations(SDEs)with general power-law and driven by a Wiener process is conducted.We prove existence of unique,strong Markovian,continuous solutions.The solutions live(a.s.)on bounded domains D=[0,K]required by applications to biology,ecology and physics with nonrandom threshold parameter K>0(i.e.the maximum carrying constant).Moreover,we present and justify nonstandard numerical methods constructed by specified balanced implicit methods(BIMs).Their weak and L^(p)-convergence follows from the fact that these methods with local Lipschitz-continuous coefficients of logistic SDEs“produce”positive numerical approximations on bounded domain[0,K](a.s.).As commonly known,standard numerical methods such as Taylor-type ones for SDEs fail to do that.Finally,asymptotic stability of nontrivial equilibria x_(∗)=K is proven for both continuous time logistic SDEs and discrete time approximations by BIMs.We exploit the technique of positive,sufficiently smooth and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.
