Korteweg-de Vries System Research Articles

Soliton molecules, bifurcation solitons and interaction solutions of a generalized (2 + 1)-dimensional korteweg-de vries system for the shallow-water waves

The central purpose of this paper is exploring the soliton molecules, bifurcation solitons and interaction solutions of the Korteweg–de Vries system based on the Hirota bilinear method. The studied system acts as an extension of the classic KdV system for the shallow-water waves, and is very useful to contribute in nonlinear wave phenomena. Firstly, the soliton molecules are obtained by adding resonance parameters in N-soliton. Then the interaction solutions between soliton/breather and soliton molecules are studied, as well as the interaction between two soliton molecules by using N-soliton. Moreover, a class of novel bifurcation solitons are derived, including Y-type bifurcation solitons, X-type bifurcation solitons and multiple-bifurcation solitons. In the end, the dynamic properties of soliton molecules, bifurcation solitons as well as the interaction solutions are presented graphically. The developed solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the generalized (2+1)-dimensional Korteweg–de Vries system better.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Extreme events in a damped Korteweg–de Vries (KdV) autonomous system: A comprehensive analysis

This manuscript explores extreme events in a three-dimensional, damped Korteweg–de Vries (KdV) autonomous system that is derived from the jerk system with a modified basin. Through rigorous stability analysis, we identify the extreme events, confirmed using phase portraits, Poincaré return maps, time series, and probability distribution functions. The Dragon-king phenomenon is obtained and self-organised criticality is discussed. The regime of extreme events in a wide parameter range was obtained through two-parameter scanning. The analog circuit was constructed to mimicking dynamics of damped KdV equation and demonstrated in real-time hardware experiment as well as PSpice simulation. The experimental results show excellent agreement with the numerically obtained results. This study is the first comprehensive examination of extreme events in the damped KdV system, highlighting their nature and potential applications in chaos-based dynamical systems.

On the study of an extended coupled KdV system: Analytical solutions and conservation laws

This paper aims to derive analytical solutions of an extended (2+1)-dimensional constant coefficients new coupled Korteweg–de Vries system. This will be achieved by implementing the classical symmetry method in conjunction with some simplest equation methods. The simplest equations that will be utilised includes among others the Bernoulli and Riccati equations. Furthermore, the conservation laws will be constructed through the multipliers approach, which subsequently reveals the conserved quantities. Moreover, a brief presentation of results obtained consisting of a variety of profile structures which include the kink type, bell and inverted bell shaped and singular wave solutions will be discussed.

Well-Posedness of the Schrödinger–Korteweg–de Vries System with Robin Boundary Conditions on the Half-Line

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball B(0,r). Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system.

Lax pair, conservation laws, breather-to-soliton transitions and modulation instability for a coupled extended modified Korteweg-de Vries system in a fluid

Lax pair, conservation laws, breather-to-soliton transitions and modulation instability for a coupled extended modified Korteweg-de Vries system in a fluid

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method.

This paper aims to extend the applications of the projected fractional improved Adomian Decomposition method (fIADM) to the fractional order new coupled Korteweg-de Vries (cKdV) system. This technique is significantly recognized for its effectiveness in addressing nonlinearities and iteratively handling fractional derivatives. The approximate solutions of the fractional-order new cKdV system are obtained by employing the improved ADM in fractional form. These solutions play a crucial role in designing and optimizing systems in engineering applications where accurate modeling of wave phenomena is essential, including fluid dynamics, plasma physics, nonlinear optics, and other mathematical physics domains. The fractional order new cKdV system, integrating fractional calculus, enhances accuracy in modeling wave interactions compared to the classical cKdV system. Comparison with exact solutions demonstrates the high accuracy and ease of application of the projected method. This proposed technique proves influential in resolving fractional coupled systems encountered in various fields, including engineering and physics. Numerical results obtained using Mathematica software further verify and demonstrate its efficacy.

Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width.

Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

Abstract In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

Soliton solutions of (2+1) complex modified Korteweg–de Vries system using improved Sardar method

Soliton solutions of (2+1) complex modified Korteweg–de Vries system using improved Sardar method

On the Lagrangian multiform structure of the extended lattice Boussinesq system

The lattice Boussinesq (lBSQ) equation is a member of the lattice Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is an infinite family of integrable systems of partial difference equations labelled by an integer $N$, where $N=2$ represents the lattice Korteweg-de Vries (KdV) system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was shown that, written as three-component system, the lBSQ system allows for extra parameters which essentially amounts to building the lattice KdV inside the lBSQ. In this paper we show that, on the level of the Lagrangian structure, this boils down to a linear combination of Lagrangians from the members of the lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding Lagrangian multiform structure is shown to exhibit a `double zero' structure.

Painlevé analysis and inelastic interactions of the lumps for a generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves

Water waves, one of the common natural phenomena, are recognized as complex and often turbulent. A generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves is conducted in this paper. We perform the Painlevé analysis and find that the system is Painlevé integrable. We study the inelastic interactions of the lumps for the system. We find that two lumps, which propagate along the curves with the equal amplitude, are symmetric about the x axis before the interaction, where x is a scaled spatial variable. After the interaction, amplitudes of the two lumps are different, but in the process of moving, the lower lump gradually increases, while the higher lump gradually decreases, and the velocities of two lumps at the infinity are equal. We observe two different inelastic interactions of the three lumps: (1) the three lumps are symmetric in time and space, and they slowly contract (before the interaction) and swell (after the interaction); (2) the three lumps slowly fuse and after the interaction they form a straight line forward, and their amplitudes are gradually equal when t → ∞.

Analysis of bifurcation, chaotic structures, lump and $ M-W $-shape soliton solutions to $ (2+1) $ complex modified Korteweg-de-Vries system

<abstract><p>This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.</p></abstract>

Higher dimensional integrable deformations of a generalized Hirota-Satsuma coupled Korteweg-de Vries system

Higher dimensional integrable deformations of a generalized Hirota-Satsuma coupled Korteweg-de Vries system

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues.

High-Performance Computational Method for an Extended Three-Coupled Korteweg–de Vries System

This paper calculates numerical solutions of an extended three-coupled Korteweg–de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation.

From decoupled integrable models to coupled ones via a deformation algorithm* *Supported by The National Natural Science Foundation (Nos. 12235007, 12090020, 11975131, 12090025).

By using a reconstruction procedure of conservation laws of different models, the deformation algorithm proposed by Lou, Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one. Using the new application to some decoupled systems such as the decoupled dispersionless Korteweg–de Vries (KdV) systems related to dispersionless waves, the decoupled KdV systems related to dispersion waves, the decoupled KdV and Burgers systems related to the linear dispersion and diffusion effects, and the decoupled KdV and Harry–Dym (HD) systems related to the linear and nonlinear dispersion effects, we have obtained various new types of higher dimensional integrable coupled systems. The new models can be used to describe the interactions among different nonlinear waves and/or different effects including the dispersionless waves (dispersionless KdV waves), the linear dispersion waves (KdV waves), the nonlinear dispersion waves (HD waves) and the diffusion effect. The method can be applied to couple all different separated integrable models.

Stabilization of the coupled ODE-linearized KdV system

This paper considers the exponential stabilization of coupled ordinary differential equation (ODE)-linearized Korteweg-de Vries (KdV) equation system coupled at right boundary point with left boundary control. Firstly, we transfer the original system into an exponentially stable target system by backstepping transformation. Secondly, we show the existence of the kernels in forward and backward transformation. Finally, we prove the exponential stability of the closed-loop system.

Dark Korteweg–De Vrise System and Its Higher-Dimensional Deformations

The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems. An arbitrary (K+1)-dimensional integrable Korteweg–de Vries (KdV) system, as an example, exhibiting symmetry, is illustrated to arise from a reconstructed deformation procedure, starting with a general symmetry integrable (1+1)-dimensional dark KdV system and its conservation laws. Physically, the dark equation systems may be related to dark matter physics. To describe nonlinear physics, both linear and nonlinear dispersions should be considered. In the original lower-dimensional integrable systems, only liner or nonlinear dispersion is included. The deformation algorithm naturally makes the model also include the linear dispersion and nonlinear dispersion.

Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method

The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.