Partial-limit Solutions and Rational Solutions with Parameter for the Fokas-Lenells Equation

A partial-limit procedure is applied to soliton solutions of the Fokas-Lenells equation. Multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop | u | 2 , the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly-changing amplitudes.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

An Improved Method for Computing Eigenpair Derivatives of Damped System

The calculation of eigenpair derivatives plays an important role in vibroengineering. This paper presents an improved algorithm for the eigenvector derivative of the damped systems by dividing it into a particular solution and general solution of the corresponding homogeneous equation. Compared with the existing methods, the proposed algorithm can significantly reduce the condition number of the equation for particular solution. Therefore, the relative errors of the calculated solutions are notably cut down. The results on two numerical examples show that such strategy is effective in reducing the condition numbers for both distinct and repeated eigenvalues.