In this paper, the global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed point theorem and a priori estimates.
We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable ...We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.展开更多
This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=ε...This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.展开更多
We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the gener...We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.展开更多
In this paper,solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows:λ^(1/2)(t)u(t,λ(t)y+x(t))=Q(y)+ε(t,y),and the estimation formula with respect toε(...In this paper,solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows:λ^(1/2)(t)u(t,λ(t)y+x(t))=Q(y)+ε(t,y),and the estimation formula with respect toε(t,y)is obtained:|ε(t,y)|≤Ca_(3)Te^(-θ)|y|+|λ^(1/2)(t)ε0|.For the CH equation,we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y→∞,and the approximation degree of the solution and Q is the same as that of initial data and Q,besides the energy distribution ofεis consistent with the distribution of the soliton Q in H^(2).展开更多
In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to...In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.展开更多
In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial s...In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.展开更多
文摘In this paper, the global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed point theorem and a priori estimates.
基金National Natural Science Foundation of China(Grants Nos.12271424 and 11871395)National Natural Science Foundation of China(Grants Nos.11971251,11631007 and 12111530003)。
文摘We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.
基金supported by two grants from the National Natural Science Foundation of China under contracts 10431060 and 10329101, respectively
文摘This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.
文摘We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.
基金supported by the National Natural Science Foundation of China (No.11371175)。
文摘In this paper,solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows:λ^(1/2)(t)u(t,λ(t)y+x(t))=Q(y)+ε(t,y),and the estimation formula with respect toε(t,y)is obtained:|ε(t,y)|≤Ca_(3)Te^(-θ)|y|+|λ^(1/2)(t)ε0|.For the CH equation,we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y→∞,and the approximation degree of the solution and Q is the same as that of initial data and Q,besides the energy distribution ofεis consistent with the distribution of the soliton Q in H^(2).
基金Yuezheng Gong’s work is partially supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202002)the Fundamental Research Funds for the Central Universities(Grant No.NS2022070)+7 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20220131)the National Natural Science Foundation of China(Grants Nos.12271252 and 12071216)Qi Hong’s work is partially supported by the National Natural Science Foundation of China(Grants No.12201297)the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems(Grant No.202001)Chunwu Wang’s work is partially supported by Science Challenge Project(Grant No.TZ2018002)National Science and Technology Major Project(J2019-II-0007-0027)Yushun Wang’s work is partially supported by the National Key Research and Development Program of China(Grant No.2018YFC1504205)the National Natural Science Foundation of China(Grants No.12171245).
文摘In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.
基金partially supported by the National Natural Science Foundation of China(12071439)the Zhejiang Provincial Natural Science Foundation of China(LY19A010016)the Natural Science Foundation of Jiangxi Province(20212BAB201016)。
文摘In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.
