Ito Equation Research Articles

Soliton solutions of weakly bound states for higher-order Ito equation

Soliton solutions of weakly bound states for higher-order Ito equation

Numerical and analytical investigations for solution of fractional Ito equation describing motions of long waves in shallow water under gravity

In this manuscript, a meshfree numerical scheme based on radial basis function-finite difference (RBF-FD) method is proposed to solve the time-fractional Ito equation. First, the temporal derivative is approximated through the finite difference technique. Then, the spatial derivatives are discretized via the multiquadric RBF. Furthermore, some useful theorems are discussed to establish the stability and convergence of the proposed numerical scheme. Finally, some test problems are presented and the results are compared with the acquired Kudryashov method and tanh method solutions to establish the computational efficiency and applicability of the proposed method.

Dynamics of closed-form invariant solutions and diversity of wave profiles of (2+1)-dimensional Ito integro-differential equation via Lie symmetry analysis

Nonlinear evolution equations are exploited in the models of complex nonlinear phenomena, explaining a number of our real-world problems in different fields of engineering and nonlinear sciences, for example, the modeling of interactions between ocean forces and the atmosphere, fiber optics, fluid dynamics, plasma physics, marine physics, and nonlinear dynamics. In this work, a comparatively new Lie symmetry analysis technique has been effectively implemented to explore some closed-form invariant solutions of the (2+1)-dimensional Ito equation. This technique is essential to minimize the number of independent variables by one at each stage until an ordinary differential equation (ODE) is formed. By utilizing the invariance analysis, the possible infinitesimal generators are derived, which are used to reduce the test problem and lead to determining the ODEs. All the exact solutions obtained are totally novel and differ significantly from previously published results. Due to the existence of some different arbitrary constants and functions, the solution provides a wealth of physical configurations. The solutions obtained in this article are thrust into the giant wave solutions such as curved shaped multi-solitons, parabolic waves, solitary waves, and traveling wave profiles displayed using three-dimensional (3D) wave patterns. Finally, we used symbolic computations in Mathematica to verify all the obtained soliton solutions. Furthermore, the current method’s superior expertise ensures that it is essentially capable of reducing the size of the computational assignment and can solve a variety of complex nonlinear partial differential equations (NLPDEs) that arise in many fields of nonlinear sciences, ocean engineering, nonlinear waves, and fluid dynamics.

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

RETRACTED: New interaction solutions to the (2 + 1)-dimensional Hirota–Satsuma–Ito equation

This study aims to determine novel analytical lump solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation through conducting symbolic computations using the Hirota direct method. This nonlinear model describes the propagation of unidirectional shallow-water waves and interactions of two long waves with different dispersion forms. Some informative descriptions of physical behavior related to the solutions obtained in this article have also been included through several 3D figures and 2D contour plots. The acquired results in this research may be beneficial for better understanding the interaction phenomena of localized nonlinear waves in different research fields of nonlinear science. It is notable that the use of computer algebra in the calculations required by the article is inevitable. Accordingly, in this work, we have used the symbolic package Mathematica. Our results will be meaningful for the investigation of the future development of lump solitons in many physical systems.

Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

A GENERALIZED METHOD AND ITS APPLICATIONS TO n-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN FRACTAL DOMAIN

A generalized local fractional Riccati differential equation (LFRDE) method, which is a combination of the local fractional Riccati differential equation method and the transformed rational function approach, is presented to obtain the non-differentiable exact traveling wave solutions for n-dimensional fractional partial differential equations. In order to verify the effectiveness of the method, the seventh-order local fractional Sawada–Kotera–Ito equation, the local fractional sine-Gordon equation and the local fractional Kadomtsev–Petviashvili equation in fractal domain are first considered. The obtained results show that the presented method involving fractal special functions, are powerful and effective for obtaining exact solutions of nonlinear fractional partial differential equations in fractal domains.

Mean exact finite-approximate contollability of linear stochastic equations in Hilbert spaces

We introduce the concepts of mean exact/mean exact finite dimensional/mean square approximate controllability in the framework of linear fractional stochastic evolution Ito equations. We show that mean square approximate controllability is equivalent to simultaneous mean exact finite dimensional and mean square approximate controllability.

DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION

<abstract> The degeneration of lump-type localized waves in the (2+1)-dimen-sional Ito equation is investigated through the parallel relationship of wave numbers. These lump-type localized waves can degenerate into three different kinds of localized wave solutions: singular lump-type localized wave, periodic variable amplitude localized wave, rogue wave. In the process of propagation, the lump-type localized waves keep the same waveform structure and amplitude. However, the periodic variable amplitude localized wave demonstrates three different kinds of waveform structures, which presents an interesting emit-absorb interaction phenomenon. By an emitting and absorbing interaction, the localized wave realizes the energy exchange from one localized wave to another, and keeps the original waveform structure. Rogue wave is a rational growing-and-decaying localized wave which is localized in both space and time. </abstract>

Applications of OHAM and MOHAM for Fractional Seventh-Order SKI Equations

In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.

Устойчивость систем линейных дифференциальных уравнений Ито с запаздываниями

The questions of instant stability of systems of linear differential equations Ito with delays based on the theory of positively reversible matrices are investigated. The ideas and methods developed by N. V. Azbelev and his students to investigate the sustainability of deterministic linear functional--differential equations are used for this. Are brought sufficient conditions $2p$--stability $(1\le p &lt;\infty)$ systems of linear differential Ito equations with delays in terms of positive reversibility of the matrices, built according to the parameters of the source system. The fulfillment of these conditions for specific equations is checked.

Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach

The paper investigates the multiple rogue wave solutions associated with the generalized Hirota–Satsuma–Ito (HSI) equation and the newly proposed extended (3 + 1)-dimensional Jimbo–Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1st, 2nd and 3rd-order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

Lie symmetry analysis, optimal system and conservation law of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Two-wave, breather wave solutions and stability analysis to the (2 + 1)-dimensional Ito equation

The current study employs the novel Hirota bilinear scheme to investigate the nonlinear model. Thus, we acquire some two-wave and breather wave solutions to the governing equation. Breathers are pulsating localized structures that have been used to mimic extreme waves in a variety of nonlinear dispersive media with a narrow banded starting process. Several recent investigations, on the other hand, imply that breathers can survive in more complex habitats, such as random seas, despite the attributed physical restrictions. The authenticity and genuineness of all the acquired solutions agreed with the original equation. In order to shed more light on these novel solutions, we plot the 3-dimensional and contour graphs to the reported solutions with some suitable values. The governing model is also stable because of the idea of linear stability. The study’s findings may help explain the physics behind some of the challenges facing ocean engineers.

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)‐dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)‐dimensional generalized Hirota–Satsuma–Ito (HSI) equation, the special fourth‐order (2+1)‐dimensional nonlinear equation, and the fifth‐order Korteweg–de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

A (2+1)-dimensional generalized Hirota–Satsuma–Ito equations: Lie symmetry analysis, invariant solutions and dynamics of soliton solutions

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov’s theorem.

Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation