The existence and stability of solitons in discrete nonlinear Schrödinger equations

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality. By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons. Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons

The existence and stability of solitons in discrete nonlinear Schrödinger equations

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality. By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons. Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons

ACE 3-001 Pemodelan Optimasi Evakuasi Tsunami di Kota Padang

Pada paper ini dibahas formulasi model optimasi sederhana untuk menentukan skenario terbaik dalam proses evakuasi tsunami. Model ini kemudian diselesaikandengan mengambil Kota Padang sebagai studi kasus. Dalam hal ini, objek observasi dibatasi pada beberapa kelurahan di Kota Padang yang dinilai memiliki dampak resiko terbesar jika terjadi tsunami. Masalah pemrograman linier yang muncul pada model diselesaikan secara numerik dengan menggunakan metode simpleks. Hasil-hasil perhitungan menunjukkan bahwa waktu evakuasi di kelurahan-kelurahan yang rawan memungkinkan kurang dari 15 menit, dengan asumsi adanya shelter tambahan yang dapat diakses oleh penduduk di Kelurahan Air Tawar Barat, Kelurahan Ulak Karang Utara, dan Kelurahan Ulak Karang Selatan. Kata kunci: Pemrograman Linier, Metode Simpleks, Model Evakuasi Tsunam

Modeling Of A Magnet-Spring System With Damping

This research discusses the modeling of a magnet-spring system by taking into account the damping factor. This magnet-spring model represents the movement of a magnet suspended from the ceiling at the bottom of a spring, and directly below it, there is another identical magnet permanently fixed on the floor. The modeling of this system is a continuation of the study by Fay and Med (2005), which discussed the same system but ignored the damping factor

Pembangkitan Pola Simetri p2mm dari Simulasi Sistem Dinamik

Pola simetri p2mm adalah wallpaper group yang mempunyai simetri rotasi , refleksi terhadap sumbu-x dan sumbu-y, dan translasi terhadap sumbu-x dan sumbu-y. Dalam artikel ini, pola simetri p2mm berwarna dibangkitkan melalui sebuah simulasi sistem dinamik diskrit menggunakan aplikasi Matlab, dengan terlebih dahulu melakukan analisis terhadap syarat dan pemilihan pada fungsi dinamiknya. Dalam pembangkitan pola p2mm ini, setiap titik pada bidang dijadikan sebagai titik awal pada iterasi sistem dinamik diskrit, dan jumlah iterasi yang dihasilkan menentukan warna pada titik tersebut. Dengan menggunakan beberapa kombinasi nilai-nilai parameter pada fungsi dinamik, diperoleh variasi pola-pola p2mm yang menarik dan estetis

Pengembangan Metode Interpolasi Splin Kubik Terapit dan Aplikasinya pada Masalah Pelacakan Trajektori Objek

Interpolasi splin kubik merupakan sebuah metode pencocokan kurva yang sangat populer karena mudah diterapkan dan menghasilkan kurva yang mulus. Pada artikel ini dibahas pengembangan metode interpolasi splin kubik untuk syarat batas terapit yang diambil dari rumus eksplisit beda hingga dengan ketelitian orde lebih tinggi. Pengembangan metode ini diterapkan pada masalah pelacakan trajektori objek (object tracking). Secara khusus, masalah ini diujikan untuk splin kubik terapit orde dua, dan hasil interpolasinya dibandingkan dengan hasil pada splin kubik alami dan splin kubik terapit orde satu. Dari simulasi data trajektori yang dibangkitkan dari kurva spiral Archimedean, diperoleh nilai galat total untuk splin kubik alami, terapit orde satu dan terapit orde dua masing-masing sebagai berikut: ,  dan . Berdasarkan hasil tersebut, disimpulkan bahwa interpolasi splin kubik terapit orde dua yang dikembangkan pada artikel ini dapat menghasilkan trajektori objek yang lebih akurat dibandingkan splin kubik alami dan splin kubik terapit orde satu. AbstrractCubic spline interpolation is a very popular curve fitting method since it is easy to implement and produces a smooth curve. This article discusses the development of the cubic spline interpolation method for a clamped boundary condition taken from finite-difference explicit formulas with higher-order accuracy. The development of this method is applied to an object tracking problem. In particular, this problem is examined for second-order clamped cubic spline, and the interpolated results are compared with those for natural and first-order clamped cubic splines. From the simulation of trajectory data generated from the Archimedean spiral curve, the total error values for natural, first-order, and second-order clamped cubic splines are respectively ,  and . Based on these results, it is concluded that the second-order clamped cubic spline interpolation developed in this article can produce a more accurate object trajectory than the natural and first-order clamped cubic splines

Stability Analysis Of The Rumor Spreading Model

The process of spreading rumors is the same as the process of spreading infectious diseases. In this case, a rumor-spreading model is discussed in this study by considering the credibility of rumors, the correlation between rumors and lives, and the classification of groups based on personality. Furthermore, the stability of the model is analyzed for the rumor-free equilibrium point and the rumor-spreading endemic equilibrium point

Analisis Solusi Persamaan Burger Sebagai Solusi Soliton Menggunakan Transformasi Hopf-Cole

Telah dilakukan penelitian untuk menganalisis solusi persamaan Burger dengan menggunakan transformasi Hopf-Cole. Penelitian ini dilatarbelakangi oleh perbedaan solusi yang diperoleh pada persamaan Burger saat mekanisme penyelesaian persamaan ini menggunakan transformasi Hopf-Cole dilandaskan pada transformasi Fourier dan separasi variabel (deret Fourier). Penelitian ini dilakukan dengan mencari solusi persamaan Burger menggunakan transformasi Hopf-Cole melalui mekanisme penyelesaian yang berlandaskan pada transformasi Fourier dan separasi variabel (deret Fourier). Berdasarkan analisis solusi soliton pada persamaan Burger, hanya mekanisme penyelesaian yang berlandaskan transformasi Fourier yang berhasil menemukan solusi soliton walaupun hanya stabil dalam selang waktu 0.1 s. Mekanisme penyelesaian yang berlandaskan separasi variabel (deret Fourier) menghasilkan solusi periodik berupa gelombang meluruh terhadap waktu.Kata kunci: deret Fourier, persamaan Burger, soliton, transformasi Hopf-Cole, transformasi Fourie

Benchmarking of the Split-Step Fourier Method on Solving a Soliton Propagation Equation in a Nonlinear Optical Medium

Benchmarking of the numerical split-step Fourier method in solving a soliton propagation equation in a nonlinear optical medium is considered. This study is carried out by comparing the solutions calculated by numerics with those obtained by analytics. In particular, the soliton propagation equation used as the object of observation is the nonlinear Schrödinger (NLS) equation, which describes optical solitons in optical fiber. By using the split-step Fourier method, we show that the split-step Fourier method is accurate. We also confirm that the nonlinear and dispersion parameters of the optical fiber influence the soliton propagation

Stability Analysis and Numerical Simulation of Prey-Predator System with Age Structure

This study developed a four-dimensional prey-predator model consisting four groups, i.e. immature prey, mature prey, immature predator, and mature predator. The reproduction of the prey depends on the mature group. Only mature predators take part in predation and prefer to consume mature prey populations. Analysis on the system by determining the equilibrium point of the prey-predator model. The results of the stability analysis obtained three equilibrium point i.e. the extinction of all species, the extinction of predator, and the existence of all species. Stability of model is analyzed based on the eigenvalues of the Jacobian matrix and by using Routh's criteria. Numerical simulations are carried out to confirm analytical results.  Furthermore, numerical simulation by different parameter is provided. There are three cases at the equilibrium point of the existence of all species where the parameters  and  are different. The results show that  the prey and predator populations will be stable or will not experience extinction for the all species equilibrium point