In this article,we consider the following coupled fractional nonlinear Schrödinger system in R^{(−Δ)su+P(x)u=μ1|u|^2p−2u+β|u|p|u|p−2u,x∈RN,(−Δ)sv+Q(x)v=μ2|v|^2p−2v+β|v|p|v|p−2v,x∈RN,u,v∈Hs(RN),where N≥2...In this article,we consider the following coupled fractional nonlinear Schrödinger system in R^{(−Δ)su+P(x)u=μ1|u|^2p−2u+β|u|p|u|p−2u,x∈RN,(−Δ)sv+Q(x)v=μ2|v|^2p−2v+β|v|p|v|p−2v,x∈RN,u,v∈Hs(RN),where N≥2,0<s<1,1<p<NN−2s,μ1>0,μ2>0 andβ∈R is a coupling constant.We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x),Q(x),p andβ.More precisely,we will show that for the attractive case,it has infinitely many non-radial positive synchronized vector solutions,and for the repulsive case,infinitely many non-radial positive segregated vector solutions can be found,where we assume that P(x)and Q(x)satisfy some algebraic decay at infinity.展开更多
We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field r...We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field related to a(x),we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem.We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.展开更多
By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are al...By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.展开更多
In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by re...In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.展开更多
基金supported by NSF of China(11701107)NSF of Guangxi Province (2017GXNSFBA198190)+1 种基金the second author is supported by NSF of China (11501143)the PhD launch scientific research projects of Guizhou Normal University (2014)
文摘In this article,we consider the following coupled fractional nonlinear Schrödinger system in R^{(−Δ)su+P(x)u=μ1|u|^2p−2u+β|u|p|u|p−2u,x∈RN,(−Δ)sv+Q(x)v=μ2|v|^2p−2v+β|v|p|v|p−2v,x∈RN,u,v∈Hs(RN),where N≥2,0<s<1,1<p<NN−2s,μ1>0,μ2>0 andβ∈R is a coupling constant.We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x),Q(x),p andβ.More precisely,we will show that for the attractive case,it has infinitely many non-radial positive synchronized vector solutions,and for the repulsive case,infinitely many non-radial positive segregated vector solutions can be found,where we assume that P(x)and Q(x)satisfy some algebraic decay at infinity.
基金National Natural Science Foundation of China(Grant Nos.11971147 and12371111)The second author is partially supported by National Natural Science Foundation of China(Grant No.11831009)+1 种基金the Fundamental Research Funds for the Central Universities(Grant Nos.KJ02072020-0319,CCNU22LJ002)The third author is supported by National Natural Science Foundation of China(Grant No.12201232)。
文摘We study the following Schr?dinger equation with variable exponent−Δu+u=u^(p+∈a)(x),u>0 in R^(N),where∈>0,1<p<N+2/N−2,a(x)∈C^(1)(R^(N))∩L^(∞)(R^(N)),N≥3.Under certain assumptions on a vector field related to a(x),we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem.We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.
基金Supported by the Nature Science Foundation of Shandong (No. 2004zx16,Q2005A01)
文摘By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
基金Supported by National Nature Science Foundation of China under Grant No.11102076Natural Science Fund for Colleges and Universities in Jiangsu Province under Grant No.15KJB110005
文摘By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.
基金Supported by National Natural Science Foundation of China under Grant No.11471174NSF of Ningbo under Grant No.2014A610018
文摘In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.
