Higher-order rational soliton solutions for the fifth-order modified KdV and KdV equations

In this paper, we construct the generalized perturbation ([Formula: see text], [Formula: see text])-fold Darboux transformation of the fifth-order modified Korteweg-de Vries (KdV) equation by the Taylor expansion. We use this transformation to derive the higher-order rational soliton solutions of the fifth-order modified KdV equation. We find that these higher-order rational solitons admit abundant interaction structures. We graphically present the dynamics behaviors from the first- to fourth-order rational solitons. Furthermore, by the Miura transformation, we obtain the complex rational soliton solutions of the fifth-order KdV equation.

Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the Chen–Lee–Liu lattice equation

Chen–Lee–Liu (CLL) lattice equation is an integrable discretization of the CLL equation which can be used to model the evolution of the self-steepening optical pulses without self-phase modulation. In this paper, the discrete N-fold Darboux transformation (DT) is used to derive the discrete kink multi-soliton solutions in terms of determinant for CLL lattice equation. Soliton fission and fusion interaction structures of such solutions are shown graphically. The details of their evolution are investigated by using numerical simulations, showing that a small noise with amplitude less than or equal to 0.01 produces a strong oscillation and instability of these kink soliton solutions. The discrete generalized perturbation [Formula: see text]-fold DT is constructed to express some rational solutions in terms of the determinants of CLL lattice equation by modifying the discrete N-fold DT. Infinitely many conservation laws for CLL lattice equation are constructed based on its Lax representation. Results in this paper might be helpful for understanding the propagation of optical pulses.

Mesoscopic perturbations of large random matrices

We consider the eigenvalues and eigenvectors of small rank perturbations of random [Formula: see text] matrices. We allow the rank of perturbation [Formula: see text] to increase with [Formula: see text], and the only assumption is [Formula: see text]. The spiked population model, proposed by Johnstone [On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29(2) (2001) 295–327], is of this kind, in which all the population eigenvalues are 1’s except for a few fixed eigenvalues. Our model is more general since we allow the number of non-unit population eigenvalues to grow with the population size. In both additive and multiplicative perturbation models, we study the nonasymptotic relation between the extreme eigenvalues of the perturbed random matrix and those of the perturbation. As [Formula: see text] goes to infinity, we derive the empirical distribution of the extreme eigenvalues of the perturbed random matrix. We also compute the appropriate projection of eigenvectors corresponding to the extreme eigenvalues of the perturbed random matrix. We prove that they are approximate eigenvectors of the perturbation. Our results can be regarded as an extension of the finite rank perturbation case to the full generality up to [Formula: see text].

Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation

Under investigation in this paper is the Kundu equation, which may be used to describe the propagation process of ultrashort optical pulses in nonlinear optics. The modulational instability of the plane-wave for the possible reason of the formation of the rogue wave (RW) is studied for the system. Based on our proposed generalized perturbation [Formula: see text]-fold Darboux transformation (DT), some new higher-order implicit RW solutions in terms of determinants are obtained by means of the generalized perturbation [Formula: see text]-fold DT, when choosing different special parameters, these results will reduce to the RW solutions of the Kaup–Newell (KN) equation, Chen–Lee–Liu (CLL) equation and Gerjikov–Ivanov (GI) equation, respectively. The relevant wave structures are shown graphically, which display abundant interesting wave structures. The dynamical behaviors and propagation stability of the first-order and second-order RW solutions are discussed by using numerical simulations, the higher-order nonlinear terms for the Kundu equation have an impact on the propagation instability of the RW. The method can also be extended to find the higher-order RW or rational solutions of other integrable nonlinear equations.

Study of improved confinement by a stepwise increase of the input heating power for tokamak plasmas

This paper is an extension of the brief study by Sarah Douglas et al. [Phys. Plasmas 20 (2013) 114504] where in the study a sinusoidal perturbation of the heating power has been studied. In this paper a stepwise increase of the heating power and its influence on the [Formula: see text]–[Formula: see text] transition are studied. Using a function, [Formula: see text] for the transition of input heating power for tokamak plasmas, i.e. the addition of the perturbation, [Formula: see text], to constant power [Formula: see text] is shown to promote the confinement, leading to the [Formula: see text]–[Formula: see text] transition at a lower value of [Formula: see text], as compared to the case of constant [Formula: see text] without the [Formula: see text] perturbation. It is seen that the input heating power [Formula: see text] that consists of constant part [Formula: see text] in addition to a function [Formula: see text] provides the [Formula: see text]–[Formula: see text] transition for relatively small [Formula: see text] and much wider range values of [Formula: see text] as compared to Sarah Douglas et al. [Phys. Plasmas 20 (2013) 114504].

The plasmonic eigenvalue problem

A plasmon of a bounded domain Ω ⊂ ℝn is a non-trivial bounded harmonic function on ℝn\∂Ω which is continuous at ∂Ω and whose exterior and interior normal derivatives at ∂Ω have a constant ratio. We call this ratio a plasmonic eigenvalue of Ω. Plasmons arise in the description of electromagnetic waves hitting a metallic particle Ω. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second-order perturbation formula. The problem can be reformulated in terms of Dirichlet–Neumann operators, and as a side result, we derive a formula for the shape derivative of these operators.

Investigations of the Isotropic G Factors for the Cubic Fe+ Centers in LiF and NaF

The g factors for the cubic Fe+centers in LiF and NaF are theoretically investigated from the perturbation formula of the g factor for an octahedral 3d7cluster including the contributions from the ligand orbital and spin-orbit coupling interactions. The increasing order of the g factor (i.e., LiF < NaF) can be ascribed to the decrease in covalency and the strength of cubic crystal-field of the systems. The validity of the results is discussed.