Ito Equation Research Articles

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.

Investigating multi-soliton patterns and dynamical characteristics of the (3+1)-dimensional equation via phase portraits

In this study, we investigate the deeper characteristics of the modified Ito equation, which can be applied across various scientific domains to represent systems influenced by noise and randomness. Multi-solitons, including 1-wave, 2-wave, and 3-wave solitons, have been successfully generated using a multiple exponential-function approach. For visual representation, the outcomes are displayed through 3D, 2D, density, and contour plots. The wave transformation is then applied to convert the studied model into an ordinary differential equation. Following this, the dynamic nature of the model is examined from various viewpoints, including bifurcation, chaotic phenomena, multistability, and sensitivity analysis. Bifurcation shows how the solution of a planar system depends on equilibrium points, and when an outward periodic force is implemented to the unperturbed planar system, it reveals chaotic characteristics. These are analyzed using tools such as 3-dimensional and 2-dimensional plots, time scale plots, and Poincaré maps. Additionally, the model’s sensitivity is assessed with varying initial values. The results underscore the effectiveness and relevance of the proposed approaches for examining solitons within a broad spectrum of nonlinear systems.

Exploring New Traveling Wave Solutions to the Nonlinear Integro-Partial Differential Equations with Stability and Modulation Instability in Industrial Engineering

In this research article, we demonstrate the generalized expansion method to investigate nonlinear integro-partial differential equations via an efficient mathematical method for generating abundant exact solutions for two types of applicable nonlinear models. Moreover, stability analysis and modulation instability are also studied for two types of nonlinear models in this present investigation. These analyses have several applications including analyzing control systems, engineering, biomedical engineering, neural networks, optical fiber communications, signal processing, nonlinear imaging techniques, oceanography, and astrophysical phenomena. To study nonlinear PDEs analytically, exact traveling wave solutions are in high demand. In this paper, the (1 + 1)-dimensional integro-differential Ito equation (IDIE), relevant in various branches of physics, statistical mechanics, condensed matter physics, quantum field theory, the dynamics of complex systems, etc., and also the (2 + 1)-dimensional integro-differential Sawda–Kotera equation (IDSKE), providing insights into the several physical fields, especially quantum gravity field theory, conformal field theory, neural networks, signal processing, control systems, etc., are investigated to obtain a variety of wave solutions in modern physics by using the mentioned method. Since abundant exact wave solutions give us vast information about the physical phenomena of the mentioned models, our analysis aims to determine various types of traveling wave solutions via a different integrable ordinary differential equation. Furthermore, the characteristics of the obtained new exact solutions have been illustrated by some figures. The method used here is candid, convenient, proficient, and overwhelming compared to other existing computational techniques in solving other current world physical problems. This article provides an exemplary practice of finding new types of analytical equations.

Demystifying the Karpelevič theorem

The statement of the Karpelevič theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevič region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito—in particular, Ito's theorem asserts that the boundary of the Karpelevič region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent.More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevič region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence providing another proof of the Karpelevič theorem.The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function λ:[0,1]⟶C such that PI(λ(α))=0, ∀α∈[0,1], where PI is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevič region are extremal whenever n>3.

Analyzing reciprocity dynamics in supply chains of public goods: a stochastic evolutionary game approach.

To start with an infinitely repeated game of supply chains of public goods, a stout reciprocity mechanism is introduced into income games to build a matric dynamic equation. The conventional evolutionary game method is employed to propose a model called the evolutionary game for the cooperative strategy of both the manufacturer and the seller groups in the supply chain of public goods. Also, white Gaussian noise (WGN) is added to reflect random interference in the evolution process. Then, a stochastic dynamic system is established, and Ito's differential equation is used to analyze both sides' strategy evolution in a game, interpret changes in system stability when random disturbance is added, and finally test the influence of different situations on the system stability by running a numerical simulation. The research shows that the stronger the reciprocity coefficient is, and the system is subjected to random interference, the greater the strategy choice change in players' decision-making procedures when the repeated game of public goods is conducted.

A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encompassing four additional second-order derivative terms: Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, and Δ6Hιι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{6}H_{\\iota \\iota }$$\\end{document}, emerging from the inclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generator using the Lie group invariance condition. Employing the conjugacy classes of the Abelian algebra, we transform the considered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.

Dynamical behaviors of various multi-solutions to the (2+1)-dimensional Ito equation

We explored the multi-solutions of the (2+1)-dimensional Ito equation based on the superposition formula. Firstly, via the superposition of exponential functions, we derived the multi-soliton wave solutions. Secondly, a new type of mixed solutions between multi-arbitrary functions and multi-kink solitons is introduced. Finally, we constructed the multi-localized wave solutions by utilizing the superposition of N-even power functions. Besides, we obtained novel interaction solutions based on the superposition formula of multi-localized wave solutions and multi-arbitrary functions. We illustrated the dynamic analysis and properties of these solutions using 3D, contour and density plots. These findings are essential for showing the propagation behavior of nonlinear waves and for explaining specific physical phenomena. They contribute significantly to the study of nonlinear waves.

Lump, periodic, travelling, semi-analytical solutions and stability analysis for the Ito integro-differential equation arising in shallow water waves

Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the (G′/G2) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

New analytical wave structures of the (3+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3+1)$$\\end{document}-dimensional extended modified Ito equation of seventh-order

Partial differential equations are frequently employed to depict issues arising across various scientific and engineering domains. Efforts have been made to analytically solve these equations, revealing shortcomings in some widely utilized methods, including modeling deficiencies and intricate solution processes. To address these limitations, diverse analytical methods have been explored. The Ito equation, introduced in 1980, underwent development, leading to the formulation of a fifth-order Ito equation. A seventh-order integrable (3+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3+1)$$\\end{document}-dimensional extended modified Ito equation emerged by augmenting this equation with three additional terms. In this study, novel exact solutions for the equation, absent in existing literature, were derived using the extended hyperbolic function and modified Kudryashov methods. To scrutinize the dynamic behavior of these findings, we presented 3D, contour, and 2D visualizations of select solutions. The results showcase numerous new solutions, underscoring the reliability and efficacy of the employed methods.

Abundant lump-type solutions of the variable-coefficient Hirota–Satsuma–Ito equation

Abundant lump-type solutions of the variable-coefficient Hirota–Satsuma–Ito equation

Nonlocal Symmetries, Consistent Riccati Expansion Solvability and Interaction Solutions of the Generalized Ito Equation

In this paper, we investigate the generalized Ito equation. By using the truncated Painlevé analysis method, we successfully derive its nonlocal symmetry and Bäcklund transformation, respectively. By introducing new dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetry. Moreover, we construct the interaction solution between soliton and cnoidal periodic wave of the equation by considering the consistent tanh expansion method. The conservation laws of the equation are also obtained with a detailed derivation.

Dynamics of transformed nonlinear waves in an extended (3+1)-dimensional Ito equation: state transitions and interactions

Dynamics of transformed nonlinear waves in an extended (3+1)-dimensional Ito equation: state transitions and interactions

Localized wave solutions and interactions of the (2+1)-dimensional Hirota–Satsuma–Ito equation

This paper studies the (2+1)-dimensional Hirota–Satsuma–Ito equation. Based on an associated Hirota bilinear form, lump-type solution, two types of interaction solutions, and breather wave solution of the (2+1)-dimensional Hirota–Satsuma–Ito equation are obtained, which are all related to the seed solution of the equation. It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons, and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton. Furthermore, the breather wave solution is also obtained by reducing the two-soliton solutions. The trajectory and period of the one-order breather wave are analyzed. The corresponding dynamical characteristics are demonstrated by the graphs.

A study of interaction soliton solutions for the $$(2+1)$$-dimensional Hirota–Satsuma–Ito equation

A study of interaction soliton solutions for the $$(2+1)$$-dimensional Hirota–Satsuma–Ito equation

Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation

Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation

On the Complexiton Solutions to the Conformable Fractional Hirota–Satsuma–Ito Equation

This study analyzes the Hirota–Satsuma–Ito equation, which discusses the propagation of unidirectional shallow‐water waves and the interactions between two long waves with different dispersion forms. For the proposed equation, the sine‐Gordon expansion method has been considered. This method is derived from the sine‐Gordon equation. Different types of solutions, namely, bright, periodic, and dark‐bright soliton solutions, are derived. When these solutions are compared to other previously published research, to our knowledge, the study concludes that they are innovative, and this method was not applied to this equation. The validation of the obtained solutions is verified and plotted as three‐dimensional figures to comprehend physical phenomena. With the proper parameter values, distinct graphs are created to convey the physical representation of specific solutions. The results of this paper show that the method effectively improves a system’s nonlinear dynamical behavior. This study will be useful to a wide range of engineers who specialize in engineering models. The findings show that the computational approach is successful, simple, and even applicable to complex systems.

Representing a second-order Ito equation as an equation with a given force structure

The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting the Rayleigh function are obtained. The obtained results are illustrated by an example of motion of a symmetric satellite in a circular orbit, assuming a change in pitch under the action of gravitational and aerodynamic forces.

Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

Degenerate behaviors and decompositions of the two-soliton solution in the (1+1)-dimensional Ito equation

In this paper, we investigate the degenerate behaviors and decompositions of two-soliton solution on the non-vanishing constant background for the (1+1)-dimensional Ito equation. By tuning the background constant, interaction phenomena of various types such as the elastic interaction of solitons, fission or fusion phenomena are presented. The degenerate two-soliton solutions which lead to a higher amplitude soliton in collisions of two solitons or describe the decay of a soliton into two new solitons are obtained. To better understand the nature of these interaction phenomena, we explore different decompositions to express the two-soliton solution as a sum of two functions. One of the function exhibits a special soliton excited by the collisions of two solitons, and the other describes the energy exchange of two solitons. Finally, under the module resonance of wave numbers, other types of the degenerate solutions of two solitons are derived and analyzed, such as the breather and rogue wave solutions.

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.