In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues an...In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.展开更多
This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a smal...This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....展开更多
The following coupled Schrodinger system with a small perturbationis considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and...The following coupled Schrodinger system with a small perturbationis considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12171171)Natural Science Foundation of Fujian Province of China(Grant Nos.2022J01303 and 2023J01121)the Scientific Research Funds of Huaqiao University。
文摘In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.
基金supported by the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province(No.S2011040000464)the Project of Department of Education of Guangdong Province(No.2012KJCX0074)+3 种基金the China Postdoctoral Science Foundation.Special Project(No.201104077)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(No.(2012)940)the Natural Fund of Zhanjiang Normal University(No.LZL1101)the Doctoral Project of Zhanjiang Normal University(No.ZL1109)
文摘This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....
基金supported by the National Natural Science Foundation of China(Nos.11126292,11201239,11371314)the Guangdong Natural Science Foundation(No.S2013010015957)the Project of Department of Education of Guangdong Province(No.2012KJCX0074)
文摘The following coupled Schrodinger system with a small perturbationis considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.
