Browsing by Author "Dr. Syed Tahir Raza Rizvi"

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Application of Bilinear Neural Network Method to Sawada-Kotera Model
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Syeda Maryam Zahra; CIIT/SP21-BSM-013/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 9882
In this work, we use the bilinear neural network method (BNNM) for solving the (2+1)- dimensional bidirectional Sawada-Kotera (bSK) equation. This model is the extension of the famous Korteweg-de Vries (KdV) equation which represents the shallow water waves on surfaces. Firstly, we use generalized Hirota bilinear approach (gHBA) and then by us- ing BNNM, we obtain exact analytical solution. BNNM is an important step toward the integration of neural networks with classic methodologies for nonlinear partial differential equations (NLPDEs), presenting both the solution in exact form and offering powerful vi- sualization of complex phenomena. This may be a method applied to a wide range of ﬁelds such as plasma, optics, acoustics, ﬂuid dynamics, and so on. BNNM models the neural network with a tensor formula and derives the exact solutions by symbolic computation. This neural network consists of layers, with their contribution in ﬁnding the solutions using activation functions and weight matrices. This method generates new test functions and in this thesis, the test functions consist of ”3-2-1” and ”3-3-1” neural network models. Here ”3-2-1” means 3 neurons in input layer, 3 neurons in ﬁrst hidden layer and a single neuron in second hidden layer. Lump, lump with one kink, rogue wave, and breather lump soliton and periodic solution for the governing model will be obtained by BNNM with the men- tioned layer. Sev
Application of CDSPM and Qualitative Analysis for Dynamical Models
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Sana Shabbir; FA22-RMT-033; Dr. Syed Tahir Raza Rizvi; LHR TP 9353
In order to understand nonlinear partial differential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many analytical methods have been pro posed to acquire solutions for NLPDEs such as Korteweg de Vries equation (KdV), Sin Gordon equation, nonlinear Schrodinger equation (NLSE), and all these equations possess ¨ solitons solutions. A soliton, which arises as a result of dispersion and nonlinearity, is a wave that retain their original features while propagating from one medium to another. The Complete Discrimination System for Polynomial Methods (CDSPM) is a collection of explicit expressions developed with the coefficients of a polynomial that contains real or symbolic variables. Liu was the first who suggested CDSPM technique. Using this method CDSPM we find both the exact, solitary wave (SW) solutions and sensitivity of nonlinear equation. Additionally, in a mathematical framework, sensitivity analysis examines to see how modifications to a system’s parameters affect its behavior. This thesis examines the complex wave patterns of the Gerjikov-Ivanov equation (GIE), commonly known as the derivative nonlinear Schrodinger equation (DNLSE) and analyt- ¨ ical solitons solutions for the cubic-quintic time-fractional nonlinear non-paraxial pulse transmission model. These modifiable model are significant because they are applied in fiber optics communication, nonlinear optics, and optical processing of signals. By using CDSPM approach we explore several solutions, such as rational, SW, and Jacobi elliptic function (JEF). The CDSPM technique is utilized to examine quasi-periodic behavior, bi furcation behavior, critical solution conditions and sensitivity analysis. We also explores the sensitivity analysis and quasi periodic behaviour of our governing models at various initial values. In addition, a number of methods for detecting quasi periodic behaviour will be addressed, including 3D and 2D graphs, time series, and Poincare maps
Application of the Bilinear Neural Network Approach to Kadomtsev-Petviashvili Model for Nonlinear Waves
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Numan Ahmed; CIIT/SP24-RMT-003/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10066
Nonlinear evolution equations (NLEEs) serve as fundamental models for wave phe-nomena in many physical systems, including fluid dynamics, plasma physics, optical pulse propagation, and atmospheric processes. They describe the spatiotemporal evolu-tion of waves in media where nonlinear interactions and dispersive effects act together. A prominent feature of such equations is the emergence of solitons, which are localized travelling waves capable of preserving their shape and speed over long distances. This remarkable property is generally attributed to a balance between nonlinear steepening and dispersive spreading. Solitons appear across a broad spectrum of physical contexts, ranging from hydrodynamics, nonlinear optics, and plasma physics to solid-state and lat-tice systems, where they often display particle-like behavior while retaining their wave nature. They have also been observed in more complex settings such as astrophysical, condensed-matter, nuclear, and other nonlinear media, highlighting their wide applica-bility and importance in both theoretical studies and practical applications of nonlinear wave dynamics. Different types of solitons exist based on their shape and behavior. Kink solitons rep-resent a transition from one state to another, similar to steps in a staircase. Breather solitons are localized waves that expand and contract over time. Rogue waves are ex-tremely large and sudden waves that can be much higher than normal waves. Lump solitons are localized in all directions and gradually disappear with distance. These d-ifferent wave structures help researchers better understand nonlinear wave behavior in real-world systems. This thesis presents a bilinear neural network modeling (BNNM) framework that cou-ples the Hirota bilinear approach with trainable neural parameters to obtain closed-form solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation
Applications of Extended Physics Informed Neural Network Method for Nonlinear Differential Equations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Muhammad Ismail (FA23-RMT-045); Dr. Syed Tahir Raza Rizvi; LHR TP 9788
Nonlinear partial differential equations featuring nonlinear terms occur in a number of engineering and physical problems, including reaction-diffusion processes, heat conduction, and population dynamics. Classical numerical techniques struggle to efficiently tackle high-dimensional problems and observe intricate nonlinear phenomena. Physics-Informed Neural Networks (PINNs) present a new opportunity in solving PDEs by incorporating physical boundary conditions inside the learning problem. Discontinuities, domain decomposition, and multi-scale phenomena, however, pose a problem to traditional PINNs. Extended PINNs (XPINNs) and conservative PINNs (cPINNs) have been proposed to overcome these drawbacks. In this thesis, we examine the use of PINN, XPINN, and cPINN in solving a reactiondiffusion equation with an exponential nonlinearity. We compare and examine their efficiency, accuracy, and stability in different parameter regimes. We will also examine the dynamics of mean square and absolute error in the given models. We will consider certain specific initial and boundary conditions to train the NN for governing model and will calculate the error of the equations by comparing the result extracted using PINN, cPINN and XPINN with the exact solution. With this, the results depict a sufficiently good agreement in the approximate solution of the governing model, the mean square error of which is of the order of 10−1 to 10−4 in the simulated equations. To the best of our knowledge, we apply this method to this model for the first time and the results are new and novel.
Applications of Variational Integrators and Solitons Solutions for Evaluation Equations
(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Sana Ghafoor; FA22-RMT-006; LHR TP 9343; Dr. Syed Tahir Raza Rizvi
n extensive variety of fields, including biology, chemistry, physics, fiber optics, mechan ics, atmospheric science, and electronics science, use nonlinear partial differential equa tions (NLPDEs), which makes them important. NLPDEs of a particular type are the non linear Schrodinger equations (NLSE). Every NLPDE that is integrable and nonlinear has ¨ a soliton solution. A specific type of solitary wave defined as a soliton has the ability to maintain its original structure even after interacting with another soliton. They are special wave packets which have the capacity to travel long distances without suffering any dis tortion. Solitons are often used in communication due to they are able to transmit signals with no errors throughout long distances and contain an abundance of data. In general, A nonlinear partial differential equation can be resolved to generate a soliton in many appli cations. Variational Integrators (VIs) is a numerical technique in which Lagrangian of the system is used in the action integral. VIs discretized the Lagrangian to obtain a discrete Euler Lagrange equation with the help of the Hamiltonian principle of stationary action. VIs are renowned for their capacity to preserve a distinct multi-symplectic structure while demonstrating desirable long-term energy characteristics. Finite-difference scheme (FDS) are a class of numerical techniques utilized by numerical analysis that approximate deriva tives by employing finite differences in order to solve differential equations (DE). During the past two decades, research on nonlinear models has frequently focused on soliton, a vital description in the NLSE analysis. In particular, the establishment of optical solitons has given rise to a theoretical basis for nonlinear optics. In this thesis, Our objective is to study the nonlinear Klein-Gordon model (NLKGM) and the Stochastic Biswas-Milovic equation (SBME) with parabolic law nonlinearity using the VIs by the use of projection technique, forward, backward, and central difference schemes and the Sub-OdE method. Utilizing the projection technique, we study the VIs of our gov erning model, NLKGM; Additionally, we investigate some of the numerical solutions by ix the use of central difference, forward, and backward techniques. The Sub-OdE approach is also utilized to obtain soliton solutions, that include avariety of solutions:three positive solitons, three Jacobian elliptic function solutions(JEFS), bright solitons, dark solitons, pe riodic solitons, rational solitons, and hyperbolic function solutions. Bose-Einstein conden sation, fiber optic sensors, plasma physics, optical communication, and other fields belong to the applications for these solitons.
Discussion on Soliton Solution via Complete Discrimination System Approach with Bifurcation Analysis
(Library Information Services COMSATS University Islamabad Lahore Campus, 2023-03-12) Bazgha Mustafa; SP22-RMT-033; Dr. Syed Tahir Raza Rizvi; LHR TP 8734
Nonlinear partial differential equations (NLPDEs) are significant because they are applied in almost every field of study, including biology, chemistry, physics, fiber optics, mechan ics, atmospheric, and electronics science. The nonlinear Schrodinger equation (NLSE) is ¨ a unique class of NLPDEs. The NLSE have applications in physical, biological, and engi neering research. Soliton is an important description in the NLSE analysis, and it has been a popular focus of research in the field of nonlinear models during the past two decades. The establishment of optical solitons, in particular, has provided a theoretical basis for the growth of nonlinear optics. Bifurcation analysis is an important tool in this process since it assists in identifying critical points, categorizing bifurcation types, and visualizing the sys tem’s behavior using bifurcation diagrams, providing substantial insights into the system’s overall dynamics and behavior. A bifurcation can also create or destroy soliton solutions, or it can affect the stability features of existing solitons. In this thesis, our objective is to illustrate the bifurcation, wave structure and topological properties of the Chiral NLSE (CNLSE) with Bohm potential (CNLSE-BP) and cubic quintic NLSE (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) by the complete discriminant system (CDS) of polynomial method (CDSPM). We study the bifurcation analysis of our governing model; bifurcation analysis is helpful in finding how systems change and exhibit various behaviors in response to changes in parameters or initial conditions. Furthermore, we also get optical solitons and wave structure like as Jacobian elliptic function (JEF), hyperbolic function, trigonometric function solutions, and rational function solution as well as also convert the JEF into solitary wave (SW) solutions. A sensi tivity study is also performed under different initial conditions. Moreover, the results offer a way to investigate optical solitons and exact solutions of pulse propagation in optical fibers.
Dynamics of a Vector-Host SIR-SI Model in Dengue Transmission
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rayan Zahid; CIIT/SP21-BSM-001/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 9877
Dengue fever is a major health problem, especially in tropical and subtropical areas, affect- ing around 50 countries. The disease is mainly spread by Aedes mosquitoes, with Aedes aegypti being the most important carrier. Urbanization, climate change, and population movement make it spread more easily. Dengue symptoms can range from mild fever to se- vere hemorrhagic fever, which can be fatal. To control the disease, effective strategies are needed, and epidemiological models are crucial for understanding disease spread, testing interventions, and predicting outbreaks. In this project, we will create a model that describes the relationship between humans and mosquitoes using a susceptible-infected-recovered (SIR) model for people and a susceptible- infected (SI) model for mosquitoes. The goal is to understand how these two groups in- teract and use this information to improve control measures. We will calculate the basic reproduction number (Ro) using the Next Generation Matrix method to determine whether the disease will spread (Ro > 1) or fade out (Ro < 1). We will also use stability analysis with the Lyapunov function to check these results. Furthermore, we will run simulations in MATLAB to test the model and explore the effects of control measures like mosquito population control, vaccination, and quarantine. By doing sensitivity analysis, we will identify the most important factors affecting disease spread and help prioritize the best in- terventions. The project aims to provide useful insights into how dengue spreads and guide effective ways to control it.
Jacobi Elliptic and Solitons for Nonlinear Schrödinger Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Marwa Ahmad; CIIT/SP17-RMT-016/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 5440
The Nonlinear Schrödinger Equation plays a central role in describing wave propagation in nonlinear media such as optical fibers, plasmas, and Bose–Einstein condensates. This study explores exact solutions of the equation using Jacobi Elliptic Functions, which provide periodic wave structures, and their limiting forms that lead to Solitons. The analysis highlights how different parameter regimes yield distinct wave profiles, ranging from periodic patterns to localized solitary waves. These solutions are important for understanding stability, modulation, and nonlinear wave dynamics in various physical systems.
Lump and Interaction Solution for the Nonlinear Schrödinger Equation
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Iram Javed; FA18-RMT-075; LHR TP 6550; Dr. Syed Tahir Raza Rizvi
Solitons are the special types of solitary waves which do not vary their shape and frequency during propagation speed and is also an attractive area of current research in nonlinear physics and mathematics as well as in fiber optics and communication technologies. There are many forms of solitons such as singular soliton, bright soliton, dark soliton and lump soliton. In this thesis, we acquire lumpsolitonsolutionsfor paraxial nonlinear Schr¨odinger equation (NLSE) with the aid of Hirota bilinear method (HBM). In space, lump soliton solutions are rationally positioned in every direction. We also get optical dromions for paraxial NLSE by using modified extended Tanh method. We get domain walls, singular dromions and combined dark-singular dromions. We also list the constraint conditions
Mathematical Modeling and Analysis of Pertussis Cough
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Syed Shan-E- Ahmad; CIIT/SP24-RMT-017/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 10078
Mathematical modeling of dynamical systems often uses ordinary differential equations to represent how key variables evolve over time. Such models support prediction, inter-pretation, and decision-making in biology, engineering, economics, and environmental science by linking mechanisms to observable outcomes. Stability analysis is central in this setting: it determines whether equilibria persist under perturbations and clarifies long-term behavior, resilience, and the potential impact of interventions. We apply these tools to a pertussis (whooping cough) model with five compartments: susceptible, latent-immunity, exposed, infectious, and recovered. The formulation in-corporates waning vaccine-induced immunity, reinfection, and disease-induced mortal-ity, and it is proven to be positive, bounded, and forward invariant. The basic repro-duction number Ro is derived via the next-generation matrix method; sensitivity analy-sis shows transmission, progression to infectivity, recovery, and vaccination rates most strongly influence persistence. Local and global stability of the disease-free and endem-ic equilibria are established. A nonstandard finite difference scheme preserves quali-tative dynamics and supports simulations. Numerical experiments confirm the analy-sis and illustrate how parameters shape outbreak magnitude and duration. Finally, an integral sliding mode control acting on vaccination robustly suppresses exposure and infection and can drive elimination under parameter uncertainty and disturbances. Keywords: Mathematical modeling; pertussis; compartmental model; Vaccination; wan-ing immunity; basic reproduction number Ro; sensitivity analysis
Numerical And Exact Solutions For Various Nonlinear
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Syed Oan Abbas Bukhari; CIIT/SP20-RMT-039/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 7641
A soliton is a particular type of solitary wave, that is retrieved as a result of integrable NLPDE. There are different type of solutions like analytic and approximate solutions exist. An exact solution is a symbolic representation of a value which solves a given equation exactly, while a numerical solution is numerical representation of a value which solves a given equation exactly or approximately. Numerical solutions are very usefull when we are unable to retrieve analytic solution. It is also used for comparison with analytic solutions. A lot of analytic and numer- ical methods exists in literature to solve the NLPDEs such as Bernouli’s equation approach, extended tanh method, modiﬁed simple equation method and RK methods etc. In this work, our purpose is to attain bell-type, kink and anti kink type, singular solitons and power series solutions for various forms of NLEEs. We’ll also obtain Weierstrass elliptic func- tions, Jacobi elliptic functions in terms of hyperbolic, trigonometric and rational functions. In this thesis, we’ll study various forms of NLEEs to attained these types of solutions with the help of various method like sine-cosine method (SCM), csch method, tan-cot method, sub-ode technique and residual power series method (RPSM). We will also compare our exact solutions with numerical solutions and then represent these solutions graphically.
Painleve Analysis and Soliton Solutions for Nonlinear Evolution Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Azeem Sakhawat; SP19-RMT-010; LHR TP 6470; Dr. Syed Tahir Raza Rizvi
Solitons are the special types of solitary waves which does not change their shape and speed during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical solitons (OS). Basically solitons are solu tions of special kind of nonlinear partial differential equation (NLPDE)and only integrable NLPDE's gives the soliton solutions. In this thesis, we study various kinds of soliton solutions of a couple of NLPDEs. We will obtain various type of solition solutions like solitary wave solutions, soliton wave solu tions, and different type of rational solution such as periodic rational and soliton rational solutions with the aid of unified method (UM). We will also investigate the integrability of our governing model by means of Painlev´e test (P-test). Any equation which satisfy the P-test shows that our governing model is resolvable with the help of inverse scattering transformation (IST). The equations which satisfy the P-test shows that these are integrable and gives the soliton solutions. There are many techniques in literature for the soliton solu tions of completely integrable NLEEs by transforming the equations into linear equations or with the aid of IST
PT-Symmetric and W-Shaped Solitons for Nonlinear Schrödinger Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ujala Safdar; FA18-RMT-035; LHR TP 6548; Dr. Syed Tahir Raza Rizvi
Solitons are the special types of solitary waves who does not change their speed and shape during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical Solitons. Optical solitons are produced due to interaction between group velocity dispersion and self-phase modulation. In this thesis, we studied the different forms of optical solitons in non-Kerr law media, parabolic, power, quadratic-cubic and anti-cubic laws with parity-time (PT) symmetric nonlinear and linear lattices. We obtain various types of soliton solutions with the as sistance of the inverse engineering scheme (IES). Under diverse physical conditions, we produce bright, dark and singular solitons. We will also obtained the dynamical behaviour of W-shaped soliton in cigar-shaped Bose Einstein condensate (BEC) in various nonlinear ities
Solitary Wave Solutions and Stability nalysis for Nonlinear Evolution Equation
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Asmavia Shahid; SP23-RMT-005; Dr. Syed Tahir Raza Rizvi; LHR TP 9587
In order to understand nonlinear partial differential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many NLPDEs such as Korteweg de Vries equation (KdV), Sin-Gordon equation, nonlinear Schr¨odinger equation (NLSE), and other equations possess solitons solutions. A soliton, which arises as a result of dispersion and nonlinearity, is a wave that retain their original features while propagating from one medium to another. The trial equation method proposed by Liu it is an advanced analyt- ical method for ﬁnding the exact solution of NLPDE in the study of solitons. Another method complete discrimination system for Polynomial Methods (CDSPM) is a collection of explicit expressions developed with the coefﬁcients of a polynomial that contains real or symbolic variables. Using this method CDSPM, we ﬁnd both the quasi periodic behaviour and sensitivity of nonlinear equation. Additionally, in a mathematical framework, sensitiv- ity analysis examines to see how modiﬁcations to a system’s parameters affect its behavior. This thesis examines nonlinear Kakutani-Matsuuchi (NKM) model of internal gravity waves. The evolution of lengthy internal gravity waves in a stratiﬁed ﬂuid medium is meticulously captured by the (1+1)-dimensional NKM model. By using trial approach we explore several solutions, such as SW, rational and Jacobi elliptic function (JEF). The CDSPM technique is utilized to examine quasi-periodic behavior, critical solution conditions, bifurcation behav- ior and sensitivity analysis. We also explores the quasi periodic behaviour and sensitivity analysis of our governing models at various initial values. Key words: Kakutani-Matsuuchi equation; solitons, complete discriminant system; quali- tative behaviour; quasi periodic behaviour; sensitivity analysis.
Soliton Dynamics, Chaos, Sensitivity and Energy Analysis to Nonlinear Models
(2024) Zunaira Iqbal and Aiman Shahbaz CIIT/SP21-BSM-034/LHR CIIT/SP21-BSM-014/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 9883
The Schwarz-Kortewegde Vries (SKdV) equation, a form of the Kortewegde Vries (KdV) equation, is a partial differential equation modeling wave propagation in shallow water. It plays a fundamental role in understanding nonlinear wave behavior, particularly soli- tonsstable, self-reinforcing waves that maintain shape and speed over long distances due to a balance between nonlinearity and dispersion. Originally applied to shallow water waves, the KdV equation has broader applications, including plasma physics, optical fiber transmission, and traffic flow modeling. A soliton is a stable, self-reinforcing wave that maintains its shape and speed over time, resulting from a balance between nonlinearity and dispersion in a medium. It is observed in systems like water waves, optical fibers, and plasma. This project investigates the Complete Discrimination System for Polynomial Methods (CDSPM), initially proposed by Liu. The trial equation technique further aids in finding solutions such as periodic, singular, kink, dark, and bright solitons by simplifying complex equations. We’ll also study the chaotic and quasi-periodic behavior of the SKdV equa- tion, using time series, phase portraits, and Poincar maps to examine solution geometries and the sensitivity analysis which reveals how small changes in initial conditions affect the system’s dynamics. We’ll also study energy balance analysis (EBA), which provides ap- proximate periodic solutions of nonlinear oscillatory systems. The method is based on the principle of energy conservation, where the total energy, combining both kinetic and po- tential energy remain invariant. Analytical soliton solutions for the SKdV model will also be explored, emphasizing their significance in applications like fiber optics communication and nonlinear optical signal processing. We’ll also study 2D/3D graphs and Poincar maps, providing insights into quasi-periodic dynamics under various initial values. Key words: Complete discriminant system; qualitative analysis; quasi periodic behaviour; sensitivity analysis; solitons.
Stationary Solutions And Optical Solitons For Nonl
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Insibat Afzal; FA17-RMT -012; Dr. Syed Tahir Raza Rizvi; LHR TP 5649
Solitons are vastly growing field of research not only in the field of nonlinear optics but they also appear in applied mathematics, telecommunication industry and in various branches of physics and biology. Specially, optical solitons arises in many different f ields including nonlinear optics. They are the light pulses that do not diffract in the mediumthrough which they passes. These days, optical solitons play a significant role in many branches of modern telecommunication mediums including internet industry, twitter, Facebook and many more. Due to the property of not changing shape and path after collision, optical solitons are used to transfer data without using repeaters. A lot of work have been done in retrieving optical and exact solutions of the solitary waves in nonlinear optics. In this dissertation, we will obtain the stationary solutions of Gerdjikov-Ivanov (GI) equation, cubic-quintic NLSE and paraxial equation by using the help of Lie group analysis. And also, we will get chirped sub-pico bright, dark and dark singular combo optical soliton solutions for Triki-Biswas equation by using some constraints. To solve Triki-Biswas equation we will use two different integrating methods namely, sine Gordon method and modified extended direct algebraic method.
Study of Breathers and M-Shaped Solutions to Nonlinear Models
(Library Information Services, CUI Lahore, 2022) Samia Ahmed; SP21-RMT-041; Dr. Syed Tahir Raza Rizvi
A soliton is a localised solitary wave with a constant structure that recovers its shape after col- lision. We obtain a soliton solution as a result of the balance dispersion and nonlinear effects. Solitons are generated in a variety of material and have several applications. Solitons have a broad range of applications, which suggests that they frequently occur in a variety of physics domains.In this thesis, exponential, hyperbolic and trigonometric function will be combined to analyse various NLEEs to get some solutions. Our goal in this work is to ﬁnd rogue soliton wave solu- tion (RSW), lump soliton wave solution (LSW), lump with one (LSW1K) and two kink soliton wave solution (LSW2K), periodic soliton waves solution (PSW), correspondence between lump kink and periodic soliton wave (LPKSW) for different types of NLEEs. Additionally, M type interaction with periodic and kink soliton waves solution (MSPKSW), M type correspondence with rogue and kink soliton waves solution (MSRKSW), M type rational soliton wave solution (MSRSW), M type rational interaction with one kink wave (MSR1K) and two kink (MSR2K), periodic cross rational soliton wave solution (PCRSW), kink cross rational soliton wave solu- tion (KCRSW), periodic cross kink soliton wave solution (PCKSW), periodic cross lump soli- ton wave solution (PCLSW), breather soliton wave (BSW), homoclinic breather soliton wave solution (HBSW) and multiwave soliton wave solutions (MWSW) will be obtained. We also observed the degeneration of waves through bilinear form and also reviewed the graphical be- haviour of our solutions for quadratic nonlinearity (QN) model and cold bosonic atoms in a zig zag optical lattice (CBAZZ) model.
Study of Chirp Solitons and Rogue Wave Solutions
(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Muhammad Sohail; LHR TP 8738; Dr. Syed Tahir Raza Rizvi
Nonlinear Evolution Equations (NLEEs) are a type of mathematical equation, commonly a partial differential equation (PDE), that describe the evolution of physical phenomena with nonlinear behaviours across time. These equations are critical in many scientific disci plines, including mathematical modelling and soliton theory. The solutions of the NLEEs represent the solitary waves (SW) called solitons. The solitary waves or solitons have a very unique history.Solitons are observed as a dispersion and nonlinearity balance. They are common in many different sectors and have intriguing qualities that make them use ful in a variety of applications. The chirped solitons are a recently popular solitary wave phenomenon. Chirp is used in spread spectrum communications as well as some sonar and radar devices. In this thesis, firstly we will obtain some chirped periodic and solitons wave by using Jaco bian elliptic function (JEF) for for higher order NLSE with anomalous dispersion regime. We also obtain some solitary waves (SW) like dark, bright, kink, hyperbolic, periodic and other solutions for the governing model. The chirp that corresponds to each of these optical solitons is also determined. We will also display the graph of our solutions in different di mensions. Secondly we will study different analytical solutions for Susceptible-Infectious Recovered (SIR) epidemic model with specific nonlinear incidence rate and spatial diffu sion like lump waves (LW), rogue waves (RW), periodic wave (PW) and periodic-cross lump waves (PCLW). This model offers useful information for containment methods by simulating and understanding the geographical spread of infectious illnesses. It advances our knowledge of how the dynamics of an epidemic are influenced by the movement of infected and susceptible people across space
Study of fractional nonlinear Schr¨odinger equation for soliton solutions
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Azrar-Ul-Hassan (CIIT/FA20-BSM-036/LHR), M. Raheel (CIIT/FA20-BSM-045/LHR); Dr. Syed Tahir Raza Rizvi; LHR TP 9905
The fact that nonlinear partial differential equations (NLPDEs) are used in practically ev- ery academic domain including biology, chemistry, physics, mechanics, ﬁber optics, at- mospheric science, and electronics research makes them signiﬁcant. One special kind of NLPDE is the nonlinear Schr¨odinger equation (NLSE), research in the ﬁelds of engineer- ing, biology, and physics has uses for the NLSE. A major subject of nonlinear model re- search during the past 20 years has been the soliton, an essential description in the NLSE analysis. Particularly, the development of optical solitons has given nonlinear optics a the- oretical foundation. In the framework of the time-fractional cubic-quintic nonlinear Schr¨odinger equation (TFCQ- NLSE), the analysis of soliton solutions is the main emphasis of this work. Wide-ranging applications of this adaptive model can be found in many complicated real phenomena, such as ﬁber optic communication, nonlinear optics, optical signal processing, laser-tissue interaction in medical science, and many others. The intensity of the cubic and quintic non- linear components determines the observed nonlinear effects, which include self-focusing, self-phase modulation, and wave-mixing. This model’s investigation employs a robust analytical approach. Our sub-ode approach, will be multitude of potential solitons and furnishes details on the interactions of laser pulses, was employed to explore this model. This inquiry yields numerous illustrated soliton solutions. Among these solutions are those that are hyperbolic, rational, singular, bright, Jacobian elliptic solutions (JES), Weierstrass elliptic solutions (WES) and periodic solutions (PS)
Study of Multiple Solitons and Nonlinear Waves
(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Saria Khizar; FA22-RMT-055; Dr. Syed Tahir Raza Rizvi; LHR TP 9345
Soliton theory is a significant topic in mathematical physics and applied mathematics with its rapid development since the 1960s. Solitons, also known as SWs, are found in the solutions of many kinds of nonlinear evolution equation (NLEE) or nonlinear partial dif ferential equations (NLPDEs). They have several unique properties and can be used to explaina number of significant physical phenomena. A soliton is a distinct type of SW that possesses the unique characteristic of maintaining its shape and velocity unchanged across collisions. The balance between the nonlinear effects and dispersion leads to a soliton so lution. Solitons are used widely in both pure and applied mathematics, particularly in areas such as differential equations, algebraic and differential geometry, Lie groups, and Lie al gebras. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epi demic outbreaks. The present thesis surveys new analytical results about the SIR model. The SIR model is one of the most basic models for describing the temporal dynamics of an infectious disease in a population. It compartmentalizes population into one of three categories: those who are Susceptible to the disease, those who are currently Infectious, and those who have Recovered (with immunity). At its most basic level, the SIR model is a set of ODEs that describes the number (or proportion) of people in each compartment over time. This thesis will analyze the SIR model and provide some solutions through the use of exponential, hyperbolic, and trigonometric functions. Our goal in this work is to find lump soliton solution (LS), rogue wave solution (RW), lump with one kink (LSK), peri odic waves solution (PW), interaction solution between lump, periodic and one kink as well as interaction solution between lump, periodic and two kink soliton wave for different types of NLPDEs. Additionally, M type interaction with periodic, M type rational soliton wave solution, periodic cross kink wave solution, breather wave, homoclinic breather wave, Kuznetsov Ma breathers, W shaped soliton, mixed type solutions will be obtained.