Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations
Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations
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DOI：
10.1007/s00205-008-0121-9
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发表时间：
2008-03
影响因子：
2.5
通讯作者：
Juncheng Wei;T. Weth
中科院分区：
文献类型：
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作者：
Juncheng Wei;T. Weth
考虑非线性椭圆方程组$$\left \{ \开始{对齐} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\，\mathbb B，\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm {in}\，\mathbb B，\\ &u，v > 0 \quad \rm{in}\，\mathbb B，\quad u=v=0 \quad \rm{on}\，\partial \mathbb B，\end{aligned} \right.$$其中和是单位球。我们证明了，对于每一个和，上述问题都有一个径向对称解（uβ，vβ），使得u β−vβ在径向变量上精确地改变符号k次。此外，当传递到一个子序列后，uβ→w+和v β→w−一致地在中，其中w =w+−w−具有精确的节域，并且是标量方程Δw−w+w3= 0在中，w= 0在上的径向对称解.在Hartree-Fock近似下，结果提供了具有强排斥的玻色-爱因斯坦双凝聚体的相分离成许多节域的理论指示.
We consider the nonlinear elliptic system $$\left \{ \begin{aligned} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\, \mathbb B,\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm{in}\, \mathbb B,\\ &u,v > 0 \quad \rm{in}\, \mathbb B,\quad u=v=0 \quad \rm{on}\, \partial \mathbb B, \end{aligned} \right.$$ whereandis the unit ball. We show that, for everyand, the above problem admits a radially symmetric solution (uβ,vβ) such thatuβ−vβchanges sign preciselyktimes in the radial variable. Furthermore, as, after passing to a subsequence,uβ→w+andvβ→w−uniformly in, wherew=w+−w−has preciselyknodal domains and is a radially symmetric solution of the scalar equation Δw−w+w3= 0 in,w= 0 on. Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.