arXiv subjects

N. Karjanto

Publications and source records attributed to N. Karjanto.

At least 55 records · Page 3Linked to original sources

arXiv2019

Soliton and breather splitting on star graphs from tricrystal Josephson junctions

We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary conditions that are determined by the continuity of the magnetic field and vanishing total fluxes. When one considers small-amplitude breather solutions, the system can be reduced into the nonlinear Schrödinger equation posed on a star graph. Using the equation, we show that a high-velocity incoming soliton is split into a transmitted component and a reflected one. The transmission is shown to be in good agreement with the transmission rate of plane waves in the linear Schrödinger equation on the same graph (i.e., a quantum graph). In the context of the sine-Gordon equation, small-amplitude breathers show similar qualitative behaviors, while large-amplitude ones produce complex dynamics.

nlin.PS↗

arXiv2019

Solitons in a chain of charge-parity-symmetric dimers

We consider an array of dual-core waveguides, which represent an optical realization of a chain of dimers, with an active (gain-loss) coupling between the cores, opposite signs of discrete diffraction in the parallel arrays, and a phase-velocity mismatch between them (which is necessary for the stability of the system). The array provides an optical emulation of the charge-parity ($\mathcal{CP}$) symmetry. The addition of the intracore cubic nonlinearity gives rise to several species of fundamental discrete solitons, which exist in continuous families, although the system is non-Hermitian. The existence and stability of the soliton families are explored by means of analytical and numerical methods. An asymptotic analysis is presented for the case of weak intersite coupling (i.e., near the anticontinuum limit), as well as weak coupling between cores in each dimer. Several families of fundamental discrete solitons are found in the semi-infinite gap of the system's spectrum, that have no counterparts in the continuum limit, as well as a branch which belongs to the finite band gap and carries over into a family of stable gap solitons in that limit. One branch develops an oscillatory instability above a critical strength of the intersite coupling, others being stable in their entire existence regions. Unlike solitons in conservative lattices, which are controlled solely by the strength of the intersite coupling, here fundamental-soliton families have several control parameters, one of which, viz., the coefficient of the intercore coupling in the active host medium, may be readily adjusted in the experiment by varying the gain applied to the medium.

nlin.PS↗

arXiv2019

On varying coefficients of spatial inhomogeneous nonlinear Schrödinger equation

A nonlinear evolution equation for wave packet surface gravity waves with variation in topography is revisited in this article. The equation is modeled by a spatial inhomogeneous nonlinear Schrödinger (NLS) equation with varying coefficients, derived by Djordjević and Redekopp (1978) and the nonlinear coefficient is later corrected by Dingemans and Otta (2001). We show analytically and qualitatively that the nonlinear coefficient and the corresponding averaging value, stated but not derived, by Benilov, Flanagan and Howlin (2005) and Benilov and Howlin (2006) are inaccurate. For a particular choice of topography and wave characteristics, the NLS equation alternates between focusing and defocusing case and hence, it does not admit the formation of a classical soliton, neither bright nor dark one.

nlin.PS↗

arXiv2019

Displaced phase-amplitude variables for waves on finite background

Wave amplification in nonlinear dispersive wave equations may be caused by nonlinear focussing of waves from a certain background. In the model of nonlinear Schrödinger equation we will introduce a transformation to displaced phase-amplitude variables with respect to a background of monochromatic waves. The potential energy in the Hamiltonian then depends essentially on the phase. Looking as a special case to phases that are time independent, the oscillator equation for the signal at each position becomes autonomous, with the change of phase with position as only driving force for a spatial evolution towards extreme waves. This is observed to be the governing process of wave amplification in classes of already known solutions of NLS, namely the Akhemediev-, Ma- and Peregrine-solitons. We investigate the case of the soliton on finite background in detail in this Letter as the solution that describes the complete spatial evolution of modulational instability from background to extreme waves.

nlin.PS↗

arXiv2019

Extreme wave phenomena in down-stream running modulated waves

Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive `extreme' waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories.

physics.flu-dyn↗

arXiv2019

Calculated threshold of supratransmission phenomena in waveguide arrays with saturable nonlinearity

In this work, we consider a semi-infinite discrete nonlinear Schrödinger equation with saturable nonlinearity driven at one edge by a driving force. The equation models the dynamics of coupled photorefractive waveguide arrays. It has been reported that when the frequency of the driving force is in the forbidden band, energy can be transmitted along the lattices provided that the driving amplitude is above a critical value. This nonlinear tunneling is called supratransmission. Here, we explain the source of supratransmission using geometric illustrations. Approximations to the critical amplitude for supratransmission are presented as well.

physics.optics↗

arXiv2018

Maximum temporal amplitude and designs of experiments for generation of extreme waves

This paper aims to describe a deterministic generation of extreme waves in a typical towing tank. Such a generation involves an input signal to be provided at the wavemaker in such a way that at a certain position in the wave tank, say at a position of a tested object, a large amplitude wave emerges. For the purpose, we consider a model called a spatial nonlinear Schrödinger equation describing the spatial propagation of a slowly varying envelope of a signal. Such a model has an exact solution known as (spatial) Soliton on a Finite Background (SFB) that is a nonlinear extension of Benjamin-Feir instability. This spatial-SFB is characterized by wave focusing leading to almost time-periodic extreme waves that appear in between phase singularities. Although phase singularities and wave focusing have been subject to a number of studies, this spatial-SFB written in the field variables has many interesting properties among which are the existence of many critical values related to the modulation length of the monochromatic signal in the far fields. These properties will be used in choosing parameters for designing experiments on extreme wave generation. In doing so, a quantity called maximum temporal amplitude (MTA) is used. This quantity measures at each location the maximum over time of the wave elevation. For a given modulation length of SFB and desired maximum amplitude at a position in a towing tank, the MTA readily shows the maximum signal that is required at the wavemaker and the amplitude amplification factor of the requested signal. Some examples of such a generation in realistic laboratory variables will be displayed.

physics.flu-dyn↗

arXiv2018

Some pros and cons of implementing parallel and block teachings for mathematics modules

The Department of Applied Mathematics at the University of Nottingham Malaysia Campus has a responsibility for delivering mathematics modules for engineering students. Due to the significantly large number of students, two methods of teaching delivery--parallel teaching and block teaching--have been implemented. This article discusses some pros and cons between these two methods, particularly for the Foundation programme and the first year of the Undergraduate programme in Engineering. Whether parallel teaching or block teaching is implemented, feedback comments from the students indicate that some areas need to be paid attention to.

math.HO↗

arXiv2018

Mollweide's formula in teaching trigonometry

Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles, Mollweide's formula is most likely to be omitted from the discussion. Mollweide's formula--which exhibits a cyclical nature--is particularly useful in checking one's result after solving an oblique triangle since all six components of the triangle are involved. It is interesting to note that proving Mollweide's formula can be performed without words. Furthermore, the Law of Tangents can be derived straightforwardly from this equation. In this article, we revisit Mollweide's formula and provide classroom examples where this equation comes into alive. In addition, we suggest that this seemingly less-known equation is to be included in the mathematics syllabus on the topic of Trigonometry.

math.HO↗

arXiv2018

English-Medium Instruction Calculus: Is flipping helpful?

This paper addresses some of the experiences encountered and some of the strategies utilized while using flipped classroom pedagogy in an undergraduate Single Variable Calculus (SVC) course in a Confucian Heritage Culture (CHC) environment. Due to issues related to instructor-student communication and students' learning styles, a theoretical framework is built upon three components for learning environments: inverted Bloom's taxonomy for educational learning objectives, English-medium instruction (EMI) and integrating technology in the classroom. In this study, four different types of instruction were designed; three classes were flipped while the fourth acts as a control. Type A means entirely flipped with instructor-created videos and integrating CAS {\sl Maxima}. Type B means entirely flipped with third-party videos. Type C is similar to Type A, but instead of the entire course being flipped, a single topic is flipped. Type D uses the traditional lecture style. By considering exam scores and students' feedback, both quantitative and qualitative aspects of the flipped pedagogy were investigated. Quantitative analysis indicates a statistically significant difference in the exam scores between instruction Type A and Type C as determined by one-way ANOVA ($F(3,306) = 2.67$, $p = 0.0477$) with small practical significance of the effect size ($η^2 = 0.0255$). Qualitative findings suggest that although students' engagement and communication with instructors have been enhanced, some challenges related to language and culture, along with minimizing competition and learning about a new technological tool remain to persist.

math.HO↗

arXiv2018

Stability of the NLS equation with viscosity effect

A nonlinear Schrödinger (NLS) equation with an effect of viscosity is derived from a Korteweg-de Vries (KdV) equation modified with viscosity using the method of multiple time-scales. It is well known that the plane-wave solution of the NLS equation exhibits modulational instability phenomenon. In this paper, the modulational instability of the plane-wave solution of the NLS equation modified with viscosity is investigated. The corresponding modulational dispersion relation is expressed as a quadratic equation with complex-valued coefficients. By restricting the modulational wavenumber into the case of narrow-banded spectra, it is observed that a type of dissipation, in this case, the effect of viscosity, stabilizes the modulational instability, as confirmed by earlier findings.

physics.flu-dyn↗

arXiv2018

Flipped classroom in Introductory Linear Algebra by utilizing Computer Algebra System {\sl SageMath} and a free electronic book

This article describes the authors' teaching experience in flipping the class of a basic undergraduate mathematics course Introductory Linear Algebra. We utilize a full-featured free electronic textbook, online lecture notes, an intranet learning management system (LMS) {\sl icampus}, the video-sharing website {\sl YouTube} and a Computer Algebra System (CAS) {\sl SageMath} in our flipped classroom approach. Prior coming to the class, the students are assigned to complete a portion of reading assignments from the free e-book and to watch recorded video lectures from {\sl YouTube} that cover the relevant materials. Announcements are made through the LMS {\sl icampus}, as well as disseminating teaching materials, assigning tasks, and fostering online communication outside the class hours. We dedicated in-class activities with problem-solving sessions, question and answer, discussion and providing instant feedback to the students. Hence, both out-of-class and in-class sessions are filled with active learning and interactive engagement. From the students' feedback and opinion, we discovered that some of them preferred the traditional classroom rather than the flipped classroom pedagogy. A possible reason is either they are used to the traditional lecture style where receiving information occurs inside the classroom or the flipped classroom approach that we implement does not satisfy students' learning styles. Nevertheless, several students enjoyed problem-solving activities where they have an opportunity to communicate with the instructors whenever they encounter some difficulty.

math.HO↗

arXiv2017

Newton's method's basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton's method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.

math.NA↗

arXiv2017

Attitude toward mathematics among the students at Nazarbayev University Foundation Year Programme

This article investigates the attitude toward mathematics among the students enrolled in the Foundation Year Programme at Nazarbayev University. The study is conducted quantitatively and an inventory developed by Tapia and Marsh II is adopted in this research. The inventory consists of 40 statements on the five-point Likert scale. Gender, specialization and final high school score in mathematics are collected. The number of valid returned questionnaires is 108. There are 55 males, 53 females, 73 Mathematical-Physics (MP), 22 Biology-Chemistry (BC) and 13 International Relations-Economics (IRE) students. Generally, they have a positive attitude toward mathematics, with the score mean and standard deviation are 3.999 and 0.531 out of five, respectively. We confirm a hypothesis on a positive correlation between previous high achievement in mathematics and favorable attitude toward it. The correlation value is $r = 0.300$, its effect size is medium and it is extremely significant ($p$-value $= 0.0008 < 0.001$). There is no significant difference between male and female students in terms of their attitude toward mathematics ($t = 0.6804$, $p$-value $= 0.2489 > 0.05$). There is a very significant difference between students who specialize in IRE and MP in terms of their attitude toward mathematics ($F(2,105) = 5.6848$, $p$-value $= 0.0045 < 0.01$).

math.HO↗

arXiv2017

Exploiting bifurcations in waveguide arrays for light detectors

An array of a finite number waveguides, driven laterally by injecting light at the outer waveguides, is considered. The array is modeled by a discrete nonlinear Schrödinger equation. It has been shown [Phys. Rev. Lett. 94, 243902 (2005)] that, when the injected light is in the proximity of a bifurcation point, such a system can be sensitive to small disturbances, making it possible to act as a light detector. Here, the optimum intensity of the injected light is discussed, and an analytical approximation is presented.

nlin.PS↗

arXiv2017

First- and second-order wave generation theory

The first-order and the second-order wave generation theory is studied in this paper. The theory is based on the fully nonlinear water wave equations. The nonlinear boundary value problem (BVP) is solved using a series expansion method. Using this method, the problem becomes a set of linear, signalling problems according to the expansion order. The first-order theory leads to a homogeneous BVP. It is a system with the first-order steering of the wavemaker motion as input and the surface wave field with propagating and evanescent modes as output. The second-order theory leads to a nonhomogeneous BVP. It is a system where the second-order steering of the wavemaker motion is prescribed in such a way that the second-order part of the surface elevation far from the wavemaker contains only the bound wave component and the free wave component vanishes. The second-order surface wave elevation consists of a superposition of bichromatic frequencies.

physics.flu-dyn↗

arXiv2017

Linear theory for single and double flap wavemakers

In this paper, we are concerned with deterministic wave generation in a hydrodynamic laboratory. A linear wavemaker theory is developed based on the fully dispersive water wave equations. The governing field equation is the Laplace equation for potential flow with several boundary conditions: the dynamic and kinematic boundary condition at the free surface, the lateral boundary condition at the wavemaker and the bottom boundary condition. In this work, we consider both single-flap and double-flap wavemakers. The velocity potential and surface wave elevation are derived, and the relation between the propagating wave height and wavemaker stroke is formulated. This formulation is then used to find how to operate the wavemaker in an efficient way to generate the desired propagating waves with minimal disturbances near the wavemaker.

physics.flu-dyn↗

arXiv2016

Interactions of bright and dark solitons with localized $\mathcal{PT}$-symmetric potentials

We study collisions of moving nonlinear-Schrödinger solitons with a $\mathcal{PT}$-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solitons are found by means of numerical simulations, and in the approximate semi-analytical form. An exact solution for the dark soliton pinned by the complex $\mathcal{PT}$-symmetric dipole is produced too.

nlin.PS↗