แนวคิดหลัก
A high-precision algorithm for solving the Gelfand-Levitan-Marchenko equation is proposed, based on the block Toeplitz Inner-Bordering method and using high-order Gregory quadrature formulas for numerical integration.
บทคัดย่อ
The paper presents a new high-precision algorithm for solving the Gelfand-Levitan-Marchenko (GLM) equation, which is crucial for the inverse nonlinear Fourier transform (NFT) problem. The algorithm is based on the block Toeplitz Inner-Bordering (TIB) method and uses high-order Gregory quadrature formulas for numerical integration.
Key highlights:
- The GLM equation is transformed into a system of linear equations with a block Toeplitz matrix structure, which can be efficiently solved using the Woodbury formula.
- High-order Gregory quadrature formulas (up to 6th order) are used to approximate the integrals in the GLM equation, achieving up to 7th order accuracy.
- Numerical experiments show that the proposed high-order schemes (G6 and G6d) outperform the second-order TIB scheme in terms of both accuracy and computational efficiency when high precision is required.
- The algorithm can be generalized to vector versions of the nonlinear Schrödinger equation, such as the Manakov equation.
The proposed method provides a significant improvement in the accuracy and efficiency of solving the inverse NFT problem compared to existing approaches.
สถิติ
The potential in the form of a chirped hyperbolic secant q(t) = A[sech(t)]^(1+iC) for A = 5.2, C = 4 is recovered.
คำพูด
"The best accuracy was provided by the schemes G6d and G6, which use the Gregory formula with 6 weight coefficients. These schemes allows one to get the sixth approximation order for the anomalous dispersion and the seventh approximation order for the normal dispersion."
"Numerical experiments have also shown that the second order TIB scheme is the most efficient on coarse grids, but when one need to get the accuracy better than 10^-4 the G6 scheme is the fastest."