论文标题
与非零边界条件的修饰基质korteweg-de vries方程的逆散射变换和孤子溶液
Inverse scattering transform and soliton solutions for the modified matrix Korteweg-de Vries equation with nonzero boundary conditions
论文作者
论文摘要
逆散射理论的发展是为了研究修改的基质Korteweg-de Vries（MMKDV）方程的初始值问题，并在Infinity的非零边界条件下，$ 200M \ tilm \ TIMES2M $ $ $ $ $ $（M \ GEQ 1）$ LAX PAIRS。在直接问题中，通过引入合适的统一转换，我们建立了适当的复杂$ z $平面，以讨论jost本征函数，散射矩阵及其方程的分析性和对称性。此外，通过wentzel-kramers-brillouin扩展，分析了逆问题中jost函数和散射矩阵的渐近行为。在反问题中，首先使用修改后的特征功能和散射系数的分析性，首先确定了MMKDV方程的普遍性Riemann-Hilbert问题。通过解决此Riemann-Hilbert问题并使用散射数据来得出潜在功能的重建公式。此外，在电势为标量和$ 2 \ times2 $对称矩阵的条件下，详细介绍了聚焦MMKDV方程的解决方案的动态行为。最后，我们提供了一些详细的证明和弱版本的痕量公式，以表明电位和散射数据的渐近阶段。
The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct problem, by introducing a suitable uniform transformation we establish the proper complex $z$-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry of the equation. Moreover the asymptotic behavior of the Jost functions and scattering matrix needed in the inverse problem are analyzed via Wentzel-Kramers-Brillouin expansion. In the inverse problem, the generalized Riemann-Hilbert problem of the mmKdV equation is first established by using the analyticity of the modified eigenfunctions and scattering coefficients. The reconstruction formula of potential function with reflection-less case is derived by solving this Riemann-Hilbert problem and using the scattering data. In addition the dynamic behavior of the solutions for the focusing mmKdV equation including one- and two- soliton solutions are presented in detail under the the condition that the potential is scalar and the $2\times2$ symmetric matrix. Finally, we provide some detailed proofs and weak version of trace formulas to show that the asymptotic phase of the potential and the scattering data.
