Nonlinear Evolution Equations Research Articles

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.

The fractional solitary wave profiles and dynamical insights with chaos analysis and sensitivity demonstration

This work investigates the dynamics of solitary waves in a thin-film ferroelectric material model, an essential component of optical systems. The study recognizes the limitations of the inverse scattering transform in solving the Cauchy problem for this conformable thin-film ferroelectric material model and applies a new Kudryashov approach and Bernoulli’s equation technique to ordinary differential equations. We apply these methods to the present model and obtain closed form analytical solutions. A variety of traveling wave and soliton solutions are constructed, including kink, bell, bright, and dark soliton solutions. It is acknowledged that the nonlinear evolution equation can be transformed into an ordinary differential equation by applying traveling wave transformations. Suitable physical parameters are chosen to create 2D, 3D, and contour plots that graphically depict the graphical dispersion of obtained fractional soliton solutions. The impact of the conformable fractional parameter is further demonstrated by the graphical representation of solitons propagation. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the system’s total energy in terms of momentum and position. While a chaos analysis describes chaotic, quasi-periodic, and periodic behaviors, a sensitivity analysis demonstrates how responsive the model is to various initial conditions. Due to their dual-purpose nature as nonlinear ferroelectric and dielectric materials, thin ferroelectric films are widely used in modern electrical devices.

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Soliton and lump and travelling wave solutions of the (3 + 1) dimensional KPB like equation with analysis of chaotic behaviors

Nonlinear evolution equations (NLEEs) have a wide range of applications in various fields, including physics, engineering, biology, economics, and more. These equations are used to describe complex systems that change over time, where the state or behavior of the system not only depends on the current situation, but may also be related to the past. A deep understanding of the solutions to NLEEs is of great significance for predicting and controlling the behavior of complex systems. The (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like (KPB-like) equation represents a partial differential equation that describes nonlinear wave phenomena in multi-dimensional spaces. This model is commonly employed in fields such as fluid dynamics, plasma physics, and nonlinear optics to study wave behaviors. This paper investigates the KPB-like equations, which have meaningful physical implications in describing nonlinear wave phenomena. Hirota bilinear method is employed to derive the bilinear form for the KPB-like equation and 1-soliton, 2-soliton, lump, and travelling wave solutions are studied to this kind of nonlinear model. In addition, the chaotic behaviors of soliton and lump solutions are explored via applying the Duffing chaotic system. To have a better understanding of dynamic structures, 3D, line, density and contour map plots of begotten results are displayed. Our results indicate the directness and effectiveness of the applied method for analyzing high-dimensional differential equations that arise in nonlinear science.

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

A novel discretization of the Yajima-Oikawa equation: Cauchy matrix approach

The Cauchy matrix approach, rooted in the Sylvester equation, plays a crucial role in defining the τ functions of nonlinear evolution equations. In this paper, the Cauchy matrix approach is employed to introduce a novel integrable semi-discrete counterpart of the one-dimensional Yajima-Oikawa (YO) system. This new system is linked with the differential-difference Kadomtsev-Petviashvili (KP) equation with self-consistent sources (SCS). Based on the Cauchy matrix approach of the KP system, we systematically construct multiple soliton and multiple pole solutions. Furthermore, we analyze and illustrate various examples of soliton solutions for further understanding.

Thin layer axion dynamo

We study interacting classical magnetic and pseudoscalar fields in frames of the axion electrodynamics. A large scale pseudoscalar field can be the coherent superposition of axions or axion like particles. We consider the evolution of these fields inside a spherical clump. Decomposing the magnetic field into the poloidal and toroidal components, we take into account their symmetry properties. Within a spherical clump, we use a thin layer approximation in the induction and Klein–Gordon equations, where the dependence of the fields on the latitude is accounted for. Then, we derive the dynamo equations in the low mode approximation. The nonlinear evolution equations for the harmonics of the magnetic and pseudoscalar fields are solved numerically. As an application, we consider a dense axion star embedded in solar plasma. The behavior of the harmonics and their typical oscillations frequencies are obtained. We suggest that such small size axionic objects, containing oscillating magnetic fields, can cause electromagnetic flashes, recently observed in the solar corona, contributing to the corona heating.

Soliton-based modeling of nano-ionic currents in transmission line

Many nonlinear evolution equations, such as the nano-ionic currents (NIC) equation, are used extensively in many scientific and technological domains particularly in nanoelectronics and bioelectronics. The mathematical modeling of NIC phenomena is vital for understanding their behavior and optimizing device performance. Our research leverages an array of mathematical methods, including multi-wave analysis, periodic wave solutions, lump soliton dynamics, breather wave phenomena, homoclinic breathers, M-shaped waveforms, and rogue wave analysis. Additionally, our investigation encompasses the exploration of single kink and double kink configurations, interactions between periodic and kink waves, interaction between M shaped with kink and rogue, interaction between M shaped with one kink, interaction between M shaped with kink and periodic, interaction between M shaped with two kinks as well as periodic wave interactions with lump waves. To further emphasize the structure of solutions derived from particular parameter choices, we include three-dimensional, two-dimensional, streamplot, and contour graphs.

Modulational instability of a pair of collinear wave trains

A coupled system of three nonlinear evolution equations is derived for a pair of collinear wave packets over finite depth fluid. The wave packets are narrow banded having different carrier wave frequencies. The evolution equations are employed to perform stability analysis of a pair of collinear wave trains. It is observed that the region of instability as well as the growth rate of instability for counter-propagating waves is greater than that for co-propagating waves when everything else remains same. The region of instability increases with the increase in the depth of the medium. It is also found that the growth rate of instability of a wave train increases with the increase in wave-steepness of the second wave train.