Linear stability analysis of pulses is considered in this review chapter. The Evans function is an analytic tool whose zeros correspond to eigenvalues. Herein, the general manner of its construction shown. Furthermore, the construction is done explicitly for the linearization of the nonlinear Schrödinger equation about the 1-soliton solution. In an explicit calculation, it is shown how the Evan...

The local stability of a nonlinear dynamical system at an equilibrium point with a pair of purely imaginary eigenvalues can be assessed through the computation of a cubic Hopf normal form coefficient, assuming the remaining eigenvalues have negative real parts. In this paper, a modal decomposition of the Hopf coefficient is proved. The decomposition provides a new methodology for analyzing the ...

Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T (λ)v = 0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG metho...

The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraic simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. For both meth...

We present an explicit analytic form for the two-breather solution of the nonlinear Schrödinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario...

Preconditioned conjugate gradient (PCG) methods have been widely used for computing a few extreme eigenvalues of large-scale linear Hermitian eigenproblems. In this paper, we study PCG methods to compute extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T (λ)v = 0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, in...

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 singulariti...

We study nonlinear electrical oscillator networks, the smallest example of which consists of a voltage-dependent capacitor, an inductor, and a resistor driven by a pure tone source. By allowing the network topology to be that of any connected graph, such circuits generalize spatially discrete nonlinear transmission lines/lattices that have proven useful in high-frequency analog devices. For suc...

We have succeeded in applying Ziglin’s test on non-integrability to nonhomogeneous nonlinear lattices (Fermi-Pasta-Ulam lattices). By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lamé type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the nonexistence of additional analyt...

We consider the quadratic eigenvalues problem (QEP) of gyroscopic systems ðlMþ lGþ KÞx 1⁄4 0, where M 1⁄4 M>;G 1⁄4 G> and K 1⁄4 K> 2 R n with M being positive definite. Guo [Numerical solution of a quadratic eigenvalue problem, Linear Algebra and its Applications 385 (2004) 391–406] showed that all eigenvalues of the QEP can be found by solving the maximal solution of a nonlinear matrix equatio...