Evans Function for Lax Operators with Algebraically Decaying Potentials
Evans Function for Lax Operators with Algebraically Decaying Potentials
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DOI：
10.1007/s00332-005-0652-7
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发表时间：
2006-02
影响因子：
3
通讯作者：
M. Klaus;D. Pelinovsky;V. Rothos
中科院分区：
文献类型：
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作者：
M. Klaus;D. Pelinovsky;V. Rothos
研究了一维可积非线性方程的代数孤子的不稳定性，这些方程包括修正的KdV方程、聚焦NLS方程、导数NLS方程和有质量的Thirring方程。我们开发的埃文斯函数，定义在相应的拉克斯运营商的代数衰减潜力的本征值的分析。标准埃文斯函数一般在本质谱中具有奇点，其可以包括具有代数衰减本征函数的嵌入本征值。我们构造了一个重正化的Evans函数，研究了当一个代数衰减的势被一个在无穷远处衰减较快的一般势扰动时，嵌入的本征值的分叉。我们发现，嵌入的特征值的分歧问题可以减少到三次或二次方程，这取决于代数潜在的衰减到零或接近一个非零常数。分叉方程的根定义对应于由不稳定代数孤子形成的非线性波的特征值。我们的结果提供了精确的信息不稳定的代数孤子的时间演化问题与可积非线性方程的转换。根据扰动的符号不同，修正KdV方程的代数孤子可以转化为传播孤子或时间周期呼吸子。代数孤子的衍生NLS和大量的Thirring方程的变换为旅行和旋转孤子的任何一个符号的扰动。最后，聚焦NLS方程的代数同宿轨道被扰动破坏，演化为时间周期的空间衰减解。
We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.