Nonlocal Reductions of the Ablowitz–Ladik Equation

Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schrödinger equation (called the Ablowitz–Ladik equation) with \(\mathcal{PT}\) symmetry. This includes the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data, and the fundamental analytic solutions. In addition, we study the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann–Hilbert problem, we derive the one- and two-soliton solutions for the nonlocal Ablowitz–Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the complete set of Jost solutions. This allows interpreting the inverse scattering transform as a generalized Fourier transform.     

Periodic processes and dispersion relations for Ablowitz–Ladik equations

There is considered the system of algebraic dispersion equations for the complexified Ablowitz–Ladik discrete equations and relevant reductions to “dark” and “white” versions, in an analogy to the NLS equation. An universal formalism presented here allows to determine effectively parameters of multiphase quasiperiodic solutions, multisoliton ones and of solutions in a form of solitons on a quasiperiodic background. This idea is connected with so-called soliton limit, which in the case of the white type solution, differs essentially from those introduced in the past and must be preceded by some modular transformation. There is a full correspondence with a similar approach for NLS equations. A few subtleties concerning solvability and examples are also discussed.     

Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method

In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann–Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then, based on the obtained Riemann–Hilbert problem, the rogue wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analysis are provided to discuss the dynamical behaviors of the rogue wave solutions and analyze how the higher-order terms affect the rogue wave.     

A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

The objective of this study was to evaluate biomathtutor by (i) investigating the impact of biomathtutor on the mathematics skills competencies of bioscience undergraduates, and (ii) assessing students’ and tutors’ reactions to biomathtutor, identifying whether and how tutors might integrate it into their curricula and blend it with more traditional teaching practices to enhance their students’ learning experiences. A multi-method approach was adopted in which a quasi-experiment and non-experimental evaluation of biomathtutor were used to collect both quantitative and qualitative data, using mathematics tests, questionnaires, tutor interviews and student focus groups. Eighty-nine bioscience undergraduates and eight tutors participated in the study. A comparison of student performance in the quasi-experiment, which adopted a pre-test-intervention-post-test methodology, revealed no significant difference between pre-test and post-test scores for either the ‘control’ group (no intervention) or for any of the mathematics learning support interventions used, including biomathtutor. Despite the limitations of the quasi-experiment which are discussed, tutors’ and their students’ reactions towards biomathtutor were very positive, with both groups agreeing that biomathtutor represents a very well designed and useful learning resource that has a valuable role to play in supporting mathematics learning within bioscience curricula. Students felt that using biomathtutor had helped them acquire new biological and mathematical knowledge and had increased their competence and confidence in mathematics, with many students confirming that they would use biomathtutor again. Tutors felt it would be useful to embed biomathtutor, where possible, into their curricula, perhaps linking it to assessment strategies or integrating it with their current more traditional teaching practices. Students indicated that they too would like to see biomathtutor embedded within their curricula, primarily because it would motivate them to use the resource. Modifications to biomathtutor, which may need to be considered in light of any potential further development of this resource, are discussed.     

Generalized double Casoratian solutions to the four-potential isospectral Ablowitz–Ladik equation

Generalized solutions in double Casoratian form of the four-potential isospectral Ablowitz–Ladik equation possessing bilinear form are derived through a matrix method for constructing double Casoratian entries. A novel class of explicit solutions, such as soliton, rational-like, Matveev, Complexiton and interaction solutions, are obtained by letting the general matrix be some special cases. Interestingly, a periodic solution is deduced from the Complexiton solution.     

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

We study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.     

On the inverse scattering problem and the low regularity solutions for the Dullin–Gottwald–Holm equation

We present an algorithm for the inverse scattering problem associated to the Dullin–Gottwald–Holm equation, arising in the study of the unidirectional propagation of waves in shallow water. In addition, a sufficient condition which guarantees the existence of the low regularity solutions for the generalized Dullin–Gottwald–Holm equation is studied by the method of energy estimation.     

Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: A unified approach

We describe a unified structure of solutions for all equations of the Ablowitz–Kaup–Newell–Segur hierarchy and their combinations. We give examples of solutions that satisfy different equations for different parameter values. In particular, we consider a rank-2 quasirational solution that can be used to investigate many integrable models in nonlinear optics. An advantage of our approach is the possibility to investigate changes in the behavior of a solution resulting from changing the model.     

Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative

We propose a (new) definition of a fractional Laplace’s transform, or Laplace’s transform of fractional order, which applies to functions which are fractional differentiable but are not differentiable, in such a manner that they cannot be analyzed by using the Djrbashian fractional derivative. After a short survey on fractional analysis based on the modified Riemann–Liouville derivative, we define the fractional Laplace’s transform. Evidence for the main properties of this fractal transformation is given, and we obtain a fractional Laplace inversion theorem.     

Generating low-discrepancy sequences from the normal distribution: Box–Muller or inverse transform?

Quasi-Monte Carlo simulation is a popular numerical method in applications, in particular, economics and finance. Since the normal distribution occurs frequently in economic and financial modeling, one often needs a method to transform low-discrepancy sequences from the uniform distribution to the normal distribution. Two well known methods used with pseudorandom numbers are the Box–Muller and the inverse transformation methods. Some researchers and financial engineers have claimed that it is incorrect to use the Box–Muller method with low-discrepancy sequences, and instead, the inverse transformation method should be used. In this paper we prove that the Box–Muller method can be used with low-discrepancy sequences, and discuss when its use could actually be advantageous. We also present numerical results that compare Box–Muller and inverse transformation methods.     

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.     

Paley–Wiener theorem for the Weinstein transform and applications

In this paper our aim is to establish the Paley–Wiener Theorem for the Weinstein Transform. Furthermore, some applications are presents. In particular some properties for the generalized translation operator associated with the Weinstein operator are proved, an integral representation and a series representation for a function in the Paley–Wiener classes are investigated.     

On the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources

A new procedure called ‘source generation’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) and its Gramm-type determinant solutions are obtained. Furthermore, the non-isospectral Pfaffianized-KP equation with self-consistent sources is constructed. This coupled system can not only be reduced to the non-isospectral Pfaffianized-KP equation, but also reduced to the non-isospectral KPESCS.     

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

Theoretical and Mathematical Physics - We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the “zeroth” time $$t_0=x$$ . It is known that their poles...