We study the following nonlinear Schrodinger system{-△u＋P（｜x｜）u=μu^3＋βv^2u,x∈R^2, -△v＋Q（｜x｜）v=υv^3＋βu^2v,x∈R^2,where P（r） and Q（r） are positive radial functions, μ〉 0, υ 〉 0, and 3 E R is a...We study the following nonlinear Schrodinger system{-△u＋P（｜x｜）u=μu^3＋βv^2u,x∈R^2, -△v＋Q（｜x｜）v=υv^3＋βu^2v,x∈R^2,where P（r） and Q（r） are positive radial functions, μ〉 0, υ 〉 0, and 3 E R is a coupling constant. This type of system arises, particularly, in models in Bose-Einstein condensates theory. Applying a finite reduction method, we construct an unbounded sequence of nonradial positive vector solutions of segregated type when β is in some suitable interval, which gives an answer to an interesting problem raised by Peng and Wang in Remark 4.1 （Arch. Ration. Mech. Anal., 208（2013）, 305-339）.展开更多
For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 su...For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 such that ifβij=βji≤b for all i≠j then there exists a radial solution(u1,u2,…uk)with uj having exactly Pj-1 zeroes.Moreover,there exists a positive constant Co such that ifβij=βji≤b(i≠j)then any solution obtained satisfies k∑i,j=1|βij|∫R^Nui^2uj^2≤C0.Therefore,the solutions exhibit a trend of phase separations asβij→-∞for i≠j.展开更多
We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three t...We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.展开更多
基金partially supported by National College Students Innovation Training Project(48)the fund from NSFC(11301204)the phD specialized grant of the Ministry of Education of China(20110144110001)
文摘We study the following nonlinear Schrodinger system{-△u＋P（｜x｜）u=μu^3＋βv^2u,x∈R^2, -△v＋Q（｜x｜）v=υv^3＋βu^2v,x∈R^2,where P（r） and Q（r） are positive radial functions, μ〉 0, υ 〉 0, and 3 E R is a coupling constant. This type of system arises, particularly, in models in Bose-Einstein condensates theory. Applying a finite reduction method, we construct an unbounded sequence of nonradial positive vector solutions of segregated type when β is in some suitable interval, which gives an answer to an interesting problem raised by Peng and Wang in Remark 4.1 （Arch. Ration. Mech. Anal., 208（2013）, 305-339）.
基金supported by the National Natural Science Foundation of China with grand numbers Nos.11671272,11331010,11771324 and 11831009
文摘For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 such that ifβij=βji≤b for all i≠j then there exists a radial solution(u1,u2,…uk)with uj having exactly Pj-1 zeroes.Moreover,there exists a positive constant Co such that ifβij=βji≤b(i≠j)then any solution obtained satisfies k∑i,j=1|βij|∫R^Nui^2uj^2≤C0.Therefore,the solutions exhibit a trend of phase separations asβij→-∞for i≠j.
文摘We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.
