Nonlinear Evolution Equations Research Articles

Thermocapillary stability of a viscoelastic liquid film falling down above or below an inclined thick wall with slip

Background:Thin viscoelastic liquid films falling down walls have been investigated theoretically since many years ago due to their applications in coating and cooling of substrates. They may also be subjected to temperature gradients and have been investigated under a variety of boundary conditions. In particular, the Navier slip boundary condition has been the subject of recent research. This condition is used when at the interface between the liquid and the wall the no-slip boundary condition does not apply due to different reasons like wall small topography, chemical coatings, etc. Methods:The small wavenumber approximation is used to derive a nonlinear evolution equation to describe the free surface deformations of the viscoelastic liquid film falling down an inclined wall. This equation is linearized and its linear stability is investigated using normal modes. The nonlinear free surface deformations are calculated numerically by means of a normal modes expansion substituted into the nonlinear evolution equation. Significant Findings:The thermocapillary stability of a thin viscoelastic film falling down a thick wall of finite thermal conductivity is investigated. Linear and nonlinear flows are examined when the interface of the liquid and the wall presents slip effects. The stability of the flow above and below (Rayleigh–Taylor) the wall is also explored. The lubrication approximation is used to derive a nonlinear evolution equation for the free surface deformation. The curves of linear growth rate, maximum growth rate and critical Marangoni number are calculated for different viscoelastic Deborah numbers. The film will be subjected to destabilizing and stabilizing Marangoni numbers. It is found that from the point of view of the linear growth rate the flow destabilizes with slip in a wavenumber range k<k+. However slip stabilizes for larger wavenumbers k>k+ up to the critical (cutoff) wavenumber. The results show that the Deborah number displaces k+ to the right. When k+ reaches the critical wavenumber by an increase of the Deborah number, slip is unable to stabilize. The corresponding critical Deborah number is derived. On the contrary, when the Deborah number is zero these slip stabilizing regions k>k+ correspond to Newtonian fluids investigated in previous works. From the point of view of the maximum growth rate slip may stabilize or destabilize increasing the slip parameter depending on the magnitude of the Marangoni, Galilei and Deborah numbers. Explicit formulas were derived for the intersections where slip may change its stability properties. Numerical solutions of the free surface nonlinear evolution equation show that slip can decrease the amplitude and may stimulate the appearance of subharmonics. The effects of different wall properties are also investigated.

An analytical investigation of the Van Der Waals gas system: Dynamics insights into bifurcation, optical pattern along with sensitivity and chaotic analysis

This paper focuses on obtaining exact solutions for the nonlinear Van der Waals gas system using the modified Khater method. Renowned as one of the latest and most precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The governing equation is transformed into an ordinary differential equation through a well-suited wave transformation. This analytical simplification makes it possible to use the provided methods to derive trigonometric, rational, and hyperbolic solutions. To illuminate the physical behavior of the model, graphical plots of selected solutions are presented. By selecting appropriate values for arbitrary factors, this visual representation enhances comprehension of the dynamical system. Furthermore, the system undergoes a certain transformation to become a planar dynamical system, and the bifurcation analysis is examined. Additionally, the sensitivity analysis of the dynamical system is conducted using the Runge–Kutta method to confirm that slight alterations in the initial conditions have minimal impact on the stability of the solution. The presence of chaotic dynamics in the Van der Waals gas system is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase profiles are used to illustrate these chaotic behaviors.

Optical soliton dynamics of the conformable nonlinear evolution equation in Bose–Einstein condensates

Optical soliton dynamics of the conformable nonlinear evolution equation in Bose–Einstein condensates

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

Nonlinear equations of evolution (NLEE) are mathematical models used in various branches of science. As a result, nonlinear equations of evolution have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear equations of evolution and solution methods. Although the origin of nonlinear equations of evolution dates back to ancient times, there have been significant developments regarding these equations from the past to the present. The main reason for this situation is that nonlinear equations of evolution involve the problem of nonlinear wave propagation. In recent years, equations of formation have become increasingly important in applied mathematics. This work focuses on the perturbation approach, often known as many scales, for nonlinear evolution equations. The article focuses on the analysis of the (1+1) dimensional third-order nonlinear Korteweg-de Vries (KdV) equation using the multiple scales method, which resulted in obtaining nonlinear Schrödinger (NLS) type equations.

An Efficient Analysis of Amplitude and Phase Dynamics in Networked MEMS-Colpitts Oscillators

Abstract This paper explores the interactions of both phase and amplitude in a network of N MEMS-Colpitts oscillators that are resistively coupled. The numerical simulations of the extensive networks of oscillators, required for emerging applications such as clock synchronization and neuromorphic computing, become computationally prohibitive as the number of oscillators increases. This complicates the design and evaluation of such systems, as understanding the effects of changes in coupling and design parameters can require many simulations. This study employs the method of multiple scales (MS) in combination with the harmonic balance method to convert the coupled differential equations governing the system of oscillators into a set of nonlinear evolution equations for the amplitude and phase of the oscillators. The amplitude and phase evolve on a timescale that is slow, commensurate with the damping in the system, compared with the fast timescale of the oscillation frequencies. The approach used in this study to describe the amplitude and phase dynamics offers significant computational efficiency (gains of 10× to 50× are shown) compared to direct numerical integration while maintaining an accurate representation of the response. The results of the presented simulations demonstrate the effect of coupling strength on the dynamics of the network, accounting for both phase and amplitude dynamics.

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

Relativistic and Heavy Ion Effects on Single‐ and Multi‐Soliton Structures of Ion‐Acoustic Waves

ABSTRACTThe fundamental characteristics of ion‐acoustic waves (IAWs) due to the relativistic effects are investigated in a collisionless unmagnetized plasma system consisting of relativistic positively charged inertial heavy ions, kappa‐distributed electrons, and Maxwellian light ions. The reductive perturbation method (RPM) is used to derive the nonlinear evolution equation like the Korteweg‐de Vries (KdV) equation. The production of single‐, double‐, and triple‐IAW solitons (IAWSs) is also investigated. It is found that for the effects of concerned parameters only rarefactive (IAWSs) are produced. The results obtained in this study are useful not only to understand the space plasma phenomena such as plasma sheet boundary layer of earth's magnetosphere, laser‐plasma interaction, but also to validate the laboratory experiments.

Analysis of the wave solutions of the nonlinear evolution equations help of advanced exp⁡(-φ(ξ))-expansion scheme

With the assistance of emblematic calculation programming, the current paper explores the specific voyaging wave arrangements from the general (2+1)- layered nonlinear development conditions by utilizing the advanced exp⁡(-φ(ξ))-expansion with time-partial boundaries. As a result, the used technique is effectively utilized and recently created some precise voyaging wave arrangements. The recently created arrangements have been communicated regarding mathematical and exaggerated capabilities. The created arrangements have been inquired into their comparing condition with the guide of the emblematic calculation programming Maple. The elements of nonlinear wave arrangements are analyzed and shown by maple18 in 3D, and 2D plots, and form plots with explicit upsides of the mind boggling boundaries are plotted. The advanced exp⁡(-φ(ξ))-expansion strategy is a solid treatment for looking through fundamental nonlinear waves that enhance an assortment of water waves in the long-frequency system that emerges in designing fields.

A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering.

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

CGC and saturation approach: Impact-parameter dependent model of perturbative QCD and combined HERA data

In this paper we confront the color glass condensate and saturation approach of Contreras [Non-linear equation in the re-summed next-to-leading order of perturbative QCD: The leading twist approximation, ] with the experimental combined HERA data and obtain its parameters. The model includes two features that are in accordance with our theoretical knowledge of deep inelastic scattering. These consist of: (i) the use of analytical solution for the nonlinear Balitsky-Kovchegov evolution equation and () the exponential behavior of the saturation momentum on the impact parameter b-dependence, characterized by Qs∝exp(−mb) which reproduces the correct behavior of the scattering amplitude at large b in accord with Froissart theorem. The model results are then compared to data at small-x for the structure function of the proton F2, the longitudinal structure function FL, the charm structure function F2cc¯, the exclusive vector meson (J/ψ,ϕ,ρ) production and deeply virtual Compton scattering. We obtain a good agreement for the processes in a wide kinematic range of Q2 at small-x. Our results provide a strong guide for finding an approach, based on color glass condensate and saturation effective theory for high-energy QCD, to make reliable predictions from first principles and for forthcoming experiments like the Electron Ion Collider and the LHeC. Published by the American Physical Society 2024

Modified homotopy approach for diffractive production in the saturation region

In this paper we continue to develop the homotopy method for solving of the nonlinear evolution equation for the diffractive production in deep inelastic scattering. We introduce part of the nonlinear corrections as a first step of this approach. This simplified nonlinear evolution equation is solved analytically taking into account the initial and boundary conditions for the process. At the second step of our approach we demonstrated that the perturbative procedure can be used for the remaining parts of the nonlinear corrections. It turns out that these corrections are small and can be estimated in the regular iterative procedure. Published by the American Physical Society 2024

On examining the predictive capabilities of two variants of the PINN in validating localized wave solutions in the generalized nonlinear Schrödinger equation

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow (x- and y-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number Re, Weber number We, angle of inclination of the plane θ and the angle of propagation of the interfacial disturbances ϕ. The temporal growth rate ωi+ and second Landau constant J2, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and β owing to the non-zero steady basic flow along the y-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of Re, We and θ, any stability that arises in two-dimensional disturbances (ϕ = 0) must also be present in three-dimensional disturbances. For ϕ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or θ) due to the involvement of B and β in the expression of J2. This explosive unstable zone vanishes after a certain value of ϕ depending upon the values of Re, We and θ, which confirms the stabilizing influence of ϕ on the thin film flow dynamics irrespective of the values of Re, We and θ.

Multiple soliton solutions and other travelling wave solutions to new structured (2+1)-dimensional integro-partial differential equation using efficient technique

The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In this study finds numerous classes of exact solutions for a new structured (2 + 1)-dimensional Ito integro-differential equation using the help of the Mathematica software. The Improved Modified Extended Tanh Function Scheme (IMETFS) is utilised to address the aforementioned equation analytically. Bright, dark, and singular soliton solutions are produced. Additionally, periodic, exponential, rational, singular periodic, and Weierstrass elliptic doubly periodic results are achieved. The method employed includes the nonlinear evolution equations that arise in a variety of real-world situations, and it is efficient, applicable, and simple to handle. For certain obtained solutions, specific options of free constants are presented in 3D, 2D, and contour graphical depictions.

New exact solutions of some (2+1)-dimensional nonlinear evolution equations and folding waves

By means of the Hirota’s bilinear method and special multi-linear variable separation ansatz, new exact solutions with low dimensional arbitrary functions of some (2+1)-dimensional nonlinear evolution equations are constructed. That is, we propose a unified method for solving the mNNV-type equations and the Burger-type equations. The key factor to the success of this method is that we have constructed some simplified Hirota’s bilinear calculation formulas in the form of variable separation of arbitrary order. Appropriate multi-valued functions are used to construct coherent structures such as the bell-type, peak-type and loop-type folding waves.