The nonlinear Schrodinger equation on the half-line
The nonlinear Schrodinger equation on the half-line
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DOI：
10.1088/0951-7715/18/4/019
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发表时间：
2005-07-01
期刊：
影响因子：
1.7
通讯作者：
Sung, LY
中科院分区：
文献类型：
--
作者：
Fokas, AS;Its, AR;Sung, LY
假设半直线上的非线性薛定谔方程的解q（x，t）存在，在Fokas（2002 Commun. Math.Phys.230 1-39），q（x，t）可以用在复k平面中公式化的矩阵Riemann-Hilbert（RH）问题的解来表示。该RH问题的跳跃矩阵具有显式的x，t依赖性，并且其根据称为谱函数的标量函数{a（k），B（k），A（k），B（k）}来定义。函数a（k）和B（k）根据g（0）（x）= q（x，0）定义，而函数A（k）和B（k）根据g（0）（t）= q（0，t）和g（1）（t）= q（x）（0，t）定义。谱函数不是独立的，但它们满足一个代数整体关系。这里我们首先证明，如果存在满足这种全局关系的谱函数，则根据上述RH问题定义的函数q（x，t）全局存在并求解非线性薛定谔方程，并且q（x，0）= q（0）（x），q（0，t）= g（0）（t）和q（x）（0，t）= g（1）（t）。然后，我们表明，适当的初始和边界条件下，它是可能的，通过解决非线性沃尔泰拉积分方程的解存在全局构造这样的谱函数。我们还表明，对于一类特定的边界条件，有可能绕过这个非线性方程和计算的谱函数只使用代数操作的全球关系;因此，对于这一类特定的边界条件，我们称之为线性化，半线上的问题可以有效地解决问题的线。可线性化边界条件的一个例子是q（x）（0，t）- rho q（0，t）= 0，其中rho是真实的常数。
Assuming that the solution q(x, t) of the nonlinear Schrodinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1-39) that q (x, t) can be represented in terms of the solution of a matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The jump matrix of this RH problem has explicit x, t dependence and it is defined in terms of the scalar functions {a(k), b(k), A(k), B(k)} referred to as spectral functions. The functions a(k) and b(k) are defined in terms of g(0)(x) = q(x, 0), while the functions A(k) and B (k) are defined in terms of g(0) (t) = q (0, t) and g(1) (t) = q(x) (0, t). The spectral functions are not independent but they satisfy an algebraic global relation. Here we first prove that if there exist spectral functions satisfying this global relation, then the function q(x, t) defined in terms of the above RH problem exists globally and solves the nonlinear Schrodinger equation, and furthermore q(x, 0) = q(0)(x), q(0, t) = g(0)(t) and q(x)(0, t) = g(1)(t). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable, the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is q(x)(0, t) - rho q(0, t) = 0 where rho is a real constant.