Higher-order integrable evolution equation and its soliton solutions

We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transformation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases: the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.A.A. and N.A. acknowledge the support of the Australian Research Council (Discovery Project DP110102068) and also thank the Volkswagen Foundation for financial support

Generalization of the Langmuir–Blodgett laws for a nonzero potential gradient

The Langmuir–Blodgett laws for cylindrical and spherical diodes and the Child–Langmuir law for planar diodes repose on the assumption that the electric field at the emission surface is zero. In the case of ion beam extraction from a plasma, the Langmuir–Blodgett relations are the typical tools of study, however, their use under the above assumption can lead to significant error in the beam distribution functions. This is because the potential gradient at the sheath/beam interface is nonzero and attains, in most practical ion beam extractors, some hundreds of kilovolts per meter. In this paper generalizations to the standard analysis of the spherical and cylindrical diodes to incorporate this difference in boundary condition are presented and the results are compared to the familiar Langmuir–Blodgett relation

Approach to first-order exact solutions of the Ablowitz-Ladik equation

We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solutions of the nonlinear Schrödinger equation (NLSE).Two of the authors (A.A. and N.A.) acknowledge the support of the Australian Research Council (Discovery Project No. DP0985394). N.A. is a grateful recipient of support from the Alexander von Humboldt Foundation (Germany)

Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation

Finite-dimensional dynamical models for solitons of the cubic-quintic complex Ginzburg-Landau equation CGLE are derived. The models describe the evolution of the pulse parameters, such as the maximum amplitude, pulse width, and chirp. A clear correspondence between attractors of the finite-dimensional dynamical systems and localized waves of the continuous dissipative system is demonstrated. It is shown that stationary solitons of the CGLE correspond to fixed points, while pulsating solitons are associated with stable limit cycles. The models show that a transformation from a stationary soliton to a pulsating soliton is the result of a Hopf bifurcation in the reduced dynamical system. The appearance of moving fronts kinks in the CGLE is related to the loss of stability of the limit cycles. Bifurcation boundaries and pulse behavior in the regions between the boundaries, for a wide range of system parameters, are found from analysis of the reduced dynamical models. We also provide a comparison between various models and their correspondence to the exact results

A theoretical reflection on the implications of the philosophy of technology for teacher education.

<p>Since the implementation of technology as relatively new school subject, challenges are constantly being posed to higher education institutions (HEIs), and in particular those engaged in teacher training and the professional development of technology teachers. Teacher training programmes had to be developed and implemented within a limited time frame in comparison to other school subjects, despite a lack of previous experience of an appropriate academic discipline, subject methodology and classroom pedagogy. Furthermore, implications on organisational and managerial level regarding its accommodation within existing structures of faculties, schools and departments at HEIs had to be accounted for. The purpose of the article was to investigate how a scientifically founded philosophical framework of technology might guide teacher training at HEIs. The following research questions served as point of departure: in which way can a scientifically founded philosophical framework of technology be indicative regarding a relevant: (1) Subject methodology of technology? (2) Underlying academic discipline for undergraduate technology education students? In answer to the first question, it was found that it is important for programme developers, coordinators and subject methodology lecturers at HEIs to acquaint themselves sufficiently with a philosophical framework for technology to direct the technology teacher’s training and professional development. It seems viable to keep subject methodology of technology autonomous, with only one lecturer responsible, and that technology education students should be conversant in the philosophical framework for technology. In answer to the second question HEIs should urgently determine the nature and composition of the relevant academic disciplines underpinning the undergraduate qualification of a specialised technology teacher. Mechanisms should also be created to forge a relationship between the academic discipline and the pedagogical content knowledge (PCK) of technology education students.</p><p> </p

Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation

We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L) equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer times. These periodic solutions comprise a family of two-parameter solutions with an arbitrary background field and a frequency of initial perturbation. The solutions are recurrent, since they return the field state to the original constant background solution after the process of nonlinear evolution has passed. These solutions can be considered as a complete resolution of the Fermi-Pasta-Ulam paradox for the A-L system. One remarkable consequence of the recurrent evolution is the nonlinear phase shift gained by the constant background wave after the process. A particular case of this family is the rational solution of the first-order or fundamental rogue wave.The authors acknowledge the support of the A.R.C. (Discovery Project DP110102068). One of the authors (N.A.) is a grateful recipient of support from the Alexander von Humboldt Foundation (Germany)

High Order Solutions and Generalized Darboux Transformations of Derivative Schr\"odinger Equation

By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Sch\"odinger equation (DNLS). These transformations are shown to lead to two solution formulas for DNLS in terms of determinants. As applications, several different types of high order solutions are calculated for this equation.Comment: 26 pages,5 figure

Multisoliton complexes in a sea of radiation modes

We derive exact analytical solutions describing multi-soliton complexes and their interactions on top of a multi-component background in media with self-focusing or self-defocusing Kerr-like nonlinearities. These results are illustrated by numerical examples which demonstrate soliton collisions and field decomposition between localized and radiation modes.Comment: 7 pages, 7 figure

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms

We present the fifth-order equation of the nonlinear Schrodinger hierarchy. This integrable partial differential ¨ equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrodinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated ¨ flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.The authors acknowledge the support of the Australian Research Council (Discovery Project No. DP140100265). N.A. and A.A. acknowledge support from the Volkswagen Stiftung and A.C. acknowledges Endeavour Postgraduate Award support