Integrable Nonlinear Differential Equations Research Articles

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

Parameter Estimation using Integral Equations

We investigate the interest of using nonlinear integral equations insteadof nonlinear differential equations in a modelisation context.In particular, we perform parameter estimation on an academic exampleusing different input-output differential and integral equations.

Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications

Abstract In this manuscript, we prove new extensions of Nashine, Wardowski, Feng-Liu, and Ćirić-type contractive inequalities using orbitally lower semi-continuous functions in an orbitally complete b b -metric space. We accomplish new multivalued common fixed point results for two families of dominated set-valued mappings in an ordered complete orbitally b b -metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral and fractional differential equations. Our results expand the hypothetical consequences of Nashine et al. (Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation, Nonlinear Anal. Model. Control. 26 (2021), no. 3, 522–533) and Rasham et al. (Common fixed point results for new Ciric-type rational multivalued-contraction with an application, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 45).

The capillary pressure curves from upscaling interfacial and unsaturated flows in porous layers with vertical heterogeneity

We provide the capillary pressure curves pc(s¯) as a function of the effective saturation s¯ based on the theoretical framework of upscaling unsaturated flows in vertically heterogeneous porous layers proposed recently (Z. Zheng, Journal of Fluid Mechanics, 950, A17, 2022). Based on the assumption of vertical gravitational-capillary equilibrium, the saturation distribution and profile shape of the invading fluid can be obtained by solving a nonlinear integral-differential equation. The capillary pressure curves pc(s¯) can then be constructed by systematically varying the injection rate. Together with the relative permeability curves k¯rn(s¯) that are already obtained. One can now provide quick estimates on the overall behaviours of interfacial and unsaturated flows in vertically-heterogeneous porous layers.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Statistical analysis for solution of non-linear integro-differential equation by using odinary and accerlated technique of Kamal-Adomian Decomposition

The integro-differential equations have a significant role in presenting the daily life phenomenon and researchers used different approaches for the solution of such problems. But it has been noticed that nonlinear integro-differential equations lack closed-form solutions. Consequently, approximate approaches are essential for determining approximate solutions to these models. The foremost goal of this research is to offer an innovative combined method by employing the Kamal transform and the Adomian decomposition method (ADM) for extracting analytical estimated, closed form, and numerical results of the non-linear differential-integral equations. The proposed method is named the Kamal Adomian Decomposition Method (KADM). To evaluate its efficiency and consistency the outcomes attained by the offered method are compared with the Laplace decomposition method which shows that our method is more efficient, that reveals the reliability of the presented method. The KADM will be used in the future to analyze new systems that emerge in many branches of science. The convergence analysis of the series of solutions is also presented. Furthermore, we provide some interesting non-trivial examples to show the validity of our main results.

Orbital b-metric spaces and related fixed point results on advanced Nashine–Wardowski–Feng–Liu type contractions with applications

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

Determining a flow structure in the region of local obstacles of different types taking into account the hydrodynamic conditions for entering the initial section

This paper investigates the influence of hydrodynamic conditions for entering the initial section of the channel located after local obstacles of various types. It is shown that the head losses in the valves and bends of pipelines and in various control elements can be several times higher than those in straight sections of the pipeline. It was established that the assumption about the rectangular shape of the velocity diagram at the entrance to the hydrodynamic initial section does not correspond to the flow pattern in real channels of technological equipment. It is proved that with the manifestation of inertia forces in the flow at the initial section of the channel, hydrodynamic energy losses usually increase, velocity and stress fields are significantly deformed. Given this, it seemed expedient to conduct a study into the processes of flow of viscous liquids in the initial section, located after local obstacles of various types. Experimental and analytical studies have confirmed that there is a significant influence of boundary conditions at the entrance to the initial section on the formation of velocity diagrams and energy loss along its length. The analytical-numerical solution to the system of differential equations describing such flow is given. While solving, the system of equations, by appropriate transformations, takes the form of a nonlinear integral-differential equation. This makes it possible to obtain correct dependences for determining the length of the velocity distribution and energy loss in the investigated section of the channel. The results of calculations of velocity fields in the region of local obstacles agree well with known ideas of the flow pattern, which is observed in physical experiments and the results of analytical solutions. The quantitative difference in results ranges within 12–20 % depending on the Reynolds number. Thus, there is reason to assert that the results of studies reported here could be the basis for devising a procedure of hydrodynamic calculation aimed at structural and operational improvement of existing and designed technological equipment.

An optimal System of Lie Subalgebras and Group-Invariant Solutions with Conserved Currents of a (3+1)-D Fifth-Order Nonlinear Model I with Applications in Electrical Electronics, Chemical Engineering and Pharmacy

AbstractHigher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3+1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as complex, topological kink and anti-kink, trigonometric and bright. We utilize the power series technique to obtain a series solution of the most complicated ordinary differential equation with forty-four terms. In addition, we reveal the dynamics of these solutions via graphical depictions. In the end, we constructed conserved currents of the underlying equation through the use of the multiplier technique. Further, we utilize the optimal system of the underlying model to derive more conserved vectors using Ibragimov’s theorem for conservation laws.

On the Solutions of the Fractional-Order Sawada–Kotera–Ito Equation and Modeling Nonlinear Structures in Fluid Mediums

This study investigates the wave solutions of the time-fractional Sawada–Kotera–Ito equation (SKIE) that arise in shallow water and many other fluid mediums by utilizing some of the most flexible and high-precision methods. The SKIE is a nonlinear integrable partial differential equation (PDE) with significant applications in shallow water dynamics and fluid mechanics. However, the traditional numerical methods used for analyzing this equation are often plagued by difficulties in handling the fractional derivatives (FDs), which lead to finding other techniques to overcome these difficulties. To address this challenge, the Adomian decomposition (AD) transform method (ADTM) and homotopy perturbation transform method (HPTM) are employed to obtain exact and numerical solutions for the time-fractional SKIE. The ADTM involves decomposing the fractional equation into a series of polynomials and solving each component iteratively. The HPTM is a modified perturbation method that uses a continuous deformation of a known solution to the desired solution. The results show that both methods can produce accurate and stable solutions for the time-fractional SKIE. In addition, we compare the numerical solutions obtained from both methods and demonstrate the superiority of the HPTM in terms of efficiency and accuracy. The study provides valuable insights into the wave solutions of shallow water dynamics and nonlinear waves in plasma, and has important implications for the study of fractional partial differential equations (FPDEs). In conclusion, the method offers effective and efficient solutions for the time-fractional SKIE and demonstrates their usefulness in solving nonlinear integrable PDEs.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper, we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces. Furthermore, we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated with and consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations. We also establish certain interesting examples to illustrate the usability of our results.

On a Non-Linear Differential-Integral Equation

On a Non-Linear Differential-Integral Equation

Using dynamic mode decomposition to predict the dynamics of a two-time non-equilibrium Green’s function

Computing the numerical solution of the Kadanoff–Baym equations, a set of nonlinear integral differential equations satisfied by the two-time Green’s functions derived from many-body perturbation theory for a quantum many-body system away from equilibrium, is a challenging task. Recently, we have successfully applied dynamic mode decomposition (DMD) to construct a data driven reduced order model that can be used to extrapolate the time-diagonal of a two-time Green’s function from numerical solutions of the KBE within a small time window. In this paper, we extend the previous work and use DMD to predict off-diagonal elements of the two-time Green’s function. We partition the two-time Green’s function into a number of one-time functions along the diagonal and subdiagonals of the two-time window as well as in horizontal and vertical directions. We use DMD to construct separate reduced order models to predict the dynamics of these one-time functions in a two-step procedure. We extrapolate along diagonal and several subdiagonals within a subdiagonal band of a two-time window in the first step. In the second step, we use DMD to extrapolate the Green’s function outside of the sub-diagonal band. We demonstrate the efficiency and accuracy of this approach by applying it to a two-band Hubbard model problem.

Suppression of Chaotic Oscillations in Small Energy Systems

The paper considers the suppression of chaotic oscillations in small energy systems that occur in emergency modes and lead to the phenomenon of voltage collapse, which corresponds to the process of voltage drop in the network, which can be accompanied by a complete shutdown of the affected area. The paper also presents a method that has been developed and that allows changing the spectrum of Lyapunov’s characteristic indicators and converting chaotic oscillations in a small power system to regular dynamic modes. The method of synthesis of control actions is based on the theorem of topological equivalence of hyperbolic nonlinear systems and their linearized models as well as on and the use of numerical integration of nonlinear differential equations describing the behavior of power systems in order to construct a phase portrait and calculate Lyapunov’s characteristic exponents. The results of the work consist in the synthesis of feedback, which ensures the formation of a spectrum of Lyapunov’s characteristic indicators with negative values. The suppression of chaotic regimes occurs by forming a spectrum of negative Lyapunov’s characteristic indicators in a closed system. The parameters of the regulator in the feedback circuit are determined using the modal control method based on the solution of the matrix algebraic Sylvester equation. The solution of the problem of transition from a chaotic regime to a regular movement in a small power system is considered. To test the operability of the proposed method of chaos suppression, the spectrum of Lyapunov’s characteristic indicators is calculated and trajectories in the phase space of the initial nonlinear system and the system with control action are constructed. For energy systems with chaotic dynamics, synthesized feedback makes it possible to suppress chaotic fluctuations and switch to regular modes, thereby preventing the occurrence of emergency modes.

On nonlinear free vibration of externally compressible fluid-loaded sandwich cylindrical shells: Curvature nonlinearity in bending and impermeability condition

A nonlinear fluid–structure interaction (FSI) model is presented for nonlinear vibration analysis of sandwich cylindrical shells subjected to an external compressible flow by considering the curvature nonlinearity in impermeability condition and bending. The sandwich shells are made of two face sheets and a central core of advanced materials including functionally graded (FG), metal foam, and anisogrid lattice composite. Based on the Kirchhoff–love hypotheses with the geometric nonlinearities in the normal strain and curvature of mid-surface, one decoupled nonlinear integral–differential equation is obtained for axisymmetric bending vibration of sandwich cylindrical shells. For the first time, the nonlinear impermeability condition is developed in order to take into account the large deformation of thin structures in FSI modeling. In both subsonic and supersonic regimes of steady irrotational compressible flow, a closed-form expression for nonlinear pressure distribution is presented according to the conservation of mass and generalized Bernoulli equation of extended potential theory. Using the Galerkin’s method in conjunction with the multiple-scale perturbation technique, the decoupled equation is solved to obtain nonlinear frequency and time–history response. Finally, parametric studies are carried out to investigate the effects of geometric ratios, boundary conditions, fluid specifications, and material variation on the responses of sandwich shells with different layups such as [Al; FG core; ZrO2], [Al; foam core; ZrO2], [Al; lattice core; Al]. Significant differences in pressure variation due to curvature nonlinearity are observed for short shells with larger vibration amplitude and larger number of half-waves in the axial direction. Therefore, the linear vibration analysis is inadequate for analyzing short sandwich cylindrical shells surrounded by an external fluid flow especially in the case of movable ends.

Constrained Nonlinear and Mixed Effects Integral Differential Equation Models for Dynamic Cell Polarity Signaling.

Polar cell growth is a process that couples the establishment of cell polarity with growth and is extremely important in the growth, development, and reproduction of eukaryotic organisms, such as pollen tube growth during plant fertilization and neuronal axon growth in animals. Pollen tube growth requires dynamic but polarized distribution and activation of a signaling protein named ROP1 to the plasma membrane via three processes: positive feedback and negative feedback regulation of ROP1 activation and its lateral diffusion along the plasma membrane. In this paper, we introduce a mechanistic integro-differential equation (IDE) along with constrained semiparametric regression to quantitatively describe the interplay among these three processes that lead to the polar distribution of active ROP1 at a steady state. Moreover, we introduce a population variability by a constrained nonlinear mixed model. Our analysis of ROP1 activity distributions from multiple pollen tubes revealed that the equilibrium between the positive and negative feedbacks for pollen tubes with similar shapes are remarkably stable, permitting us to infer an inherent quantitative relationship between the positive and negative feedback loops that defines the tip growth of pollen tubes and the polarity of tip growth.

Fixed Point Theorems of Superlinear Operators with Applications

In this paper, by using the partial order method and monotone iterative techniques, the existence and uniqueness of fixed points for a class of superlinear operators are studied, without requiring any compactness or continuity. As corollaries, the new fixed point theorems for α -convex operators α > 1 , e -convex operators, positive α homogeneous operator α > 1 , generalized e -convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.

A new nonlinear integral-differential equation describing Rossby waves and its related properties

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.