Caudrey Dodd Gibbon Kotera Sawada Equation Research Articles

Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

Numerical study of nonlinear time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising in propagation of waves

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab.

Decomposition and linear superposition of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Generally, the principle of linear superposition is not applicable in nonlinear systems. The results of our researches prove that for some certain types of decomposed solutions of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation, the linear superposition principle holds. First, beginning with the potential (2+1)-dimensional CDGKS equation, six sets of decomposition equations are formulated by the formally variable separation approach. On the basis of these decompositions, the linear superposition solutions are then discussed. More specifically, the combination of two periodic wave solutions, soliton and periodic wave interaction solution and the multi-soliton solution are obtained.

Variable-coefficient polynomial function method for finding the lump-type solutions of integrable system with variable coefficients

In this work, we study a ([Formula: see text])-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. The new lump and lump-soliton solutions are obtained by the variable-coefficient polynomial function method. We used 3D graphs, contour plots and density graphs to show that the amplitude and velocity of solitons are affected by some variable coefficients. It is proved that the polynomial function method with variable coefficients is very direct and effective for solving lump-type solutions in variable coefficient integrable systems, and more new conclusions can be obtained.

Solving nonlinear soliton equations using improved physics-informed neural networks with adaptive mechanisms

Partial differential equations (PDEs) are important tools for scientific research and are widely used in various fields. However, it is usually very difficult to obtain accurate analytical solutions of PDEs, and numerical methods to solve PDEs are often computationally intensive and very time-consuming. In recent years, Physics Informed Neural Networks (PINNs) have been successfully applied to find numerical solutions of PDEs and have shown great potential. All the while, solitary waves have been of great interest to researchers in the field of nonlinear science. In this paper, we perform numerical simulations of solitary wave solutions of several PDEs using improved PINNs. The improved PINNs not only incorporate constraints on the control equations to ensure the interpretability of the prediction results, which is important for physical field simulations, in addition, an adaptive activation function is introduced. By introducing hyperparameters in the activation function to change the slope of the activation function to avoid the disappearance of the gradient, computing time is saved thereby speeding up training. In this paper, the mKdV equation, the improved Boussinesq equation, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the p-gBKP equation are selected for study, and the errors of the simulation results are analyzed to assess the accuracy of the predicted solitary wave solution. The experimental results show that the improved PINNs are significantly better than the traditional PINNs with shorter training time but more accurate prediction results. The improved PINNs improve the training speed by more than 1.5 times compared with the traditional PINNs, while maintaining the prediction error less than 10−2 in this order of magnitude.

Spectral stability of multi-solitons for generalized Hamiltonian system I: The Caudrey–Dodd–Gibbon–Sawada–Kotera equation

We consider a generalized Hamiltonian system called the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, which is a natural extension of the Korteweg–de Vries (KdV) equation. Our main results are two-fold. Firstly, spectral stability of solitons and multi-solitons are presented by employing some inverse scattering transform techniques. The main difficulty arises from the third order eigenvalue problem of the equation. The second result is a Hamilton–Krein index identity, which verifies a stability criterion established by Maddocks and Sachs (1993) in showing the Lyapunov stability of KdV N-solitons. However, due to generalized Hamiltonian system nature of the CDGSK equation, it does not possess the L2 conservation law. The linearized operators around the multi-solitons have no spectra gap any more. The main ingredient of the proof is new operator identities for higher order linearized Hamiltonians motivated by the Sylvester Law of Inertia. Such operator identities are shown by employing the recursion operators of the CDGSK equation.

Towering and internal rogue waves induced by two-layer interaction in non-uniform fluid. A 2D non-autonomous gCDGKSE

A Generalized (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation (2D- gCDGKSE) is an integro-differential equation that describes tow-layer fluid interaction. The non-autonomous (2+1)-dimensional gCDGKSE (NAUT-gCDGKSE) was rarely considered in the literature. In the previous works, the concepts of two-layer fluid interaction and non-uniform fluid were not explored. This motivated us to focus the attention on these themes. Our objective is to inspecting waves structures in non-uniform fluid which describes fluid flows near a solid boundary. Thus, the present work is completely new. Our objective, here, is to inspect waves which are similar to those created in waterfall, water waves behind dams, boat sailing, in the network of canals during water release, and internal waves in submarine. In a uniform fluid, rogue waves occur in open oceans and seas, while in the present case of non-uniform fluid, towering and internal rogue waves occur near barriers (islands) and near submarine, respectively. This was consolidated experimentally, as it was shown that rogue wave is produced in a water tank (which is with solid boundary). The exact solutions of NAUT-gCDGKSE are derived here, by implementing the extended unified method (EUM). In applications, it is found that the EUM is of lower time cost in symbolic computation, than when using Lie symmetry, Darboux and AutoBucklund transformations. The results obtained here are evaluated numerically, and they are displayed in graphs. They reveal multiple waves structures with relevance to waves created near a solid boundary. Among them are towering and internal rogue waves, internal (hollowed) and bulge-U-shape wave and S-shape wave, water fall, saddle wave, and dromoions.

Fractional derivative-based performance analysis to Caudrey–Dodd–Gibbon–Sawada–Kotera equation

This paper explores the innovative soliton solutions to the fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation with Beta and Atangana–Baleanu (AB) fractional derivatives. The new auxiliary equation method (NAEM) is used to extricate novel analytical solutions to the CDGSK equation. A comparative analysis of the solutions is conducted by implementing these fractional derivatives. The impact of different values of fractional parameter on the behavior of obtained solutions is also explored. The shapes and dynamics of the retrieved solutions are portrayed employing 3D and 2D graphs. These new solitary wave solutions are shown to encompass dark, bright, periodic, and singular wave solutions. Analytical solutions expound that this method is an effective mathematical tool to find exact traveling wave solutions to nonlinear models found in several fields of science and engineering.

Resonance Y-shape solitons and mixed solutions for a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Under the well-known bilinear method of Hirota, the specific expression for N-soliton solutions of (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on N-soliton solutions, resonant Y-type and X-type soliton solutions are generated. Under the new proposed constraint, combined with the velocity resonance method and module resonant method, the mixed solutions of resonant Y-type solitons and line waves and breather solutions are found. Finally, with the support of long-wave limit method, the interaction between resonant Y-type solitons and higher-order lumps is shown, and the motion trajectory equation before and after the interaction between lumps and resonant Y-type solitons is derived. These new results greatly extend the exact solution of (2+1)-dimensional gCDGKS equation already available in the literature and provide new ideas for studying the dynamical behaviors of fluid mechanic, soliton and shallow water wave and so on.

N‐Lump to the (2+1)‐Dimensional Variable‐Coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

In this research, the (2 + 1)‐dimensional (D) variable‐coefficient (VC) Caudrey–Dodd–Gibbon–Kotera–Sawada model used in soliton hypothesis and implemented by operating the Hirota bilinear scheme is studied. A few modern exact analytical outcomes containing interaction between a lump‐two kink soliton, interaction between two‐lump, the interaction between two‐lump soliton, lump‐periodic, and lump‐three kink outcomes for the (2 + 1)‐D VC Caudrey–Dodd–Gibbon–Kotera–Sawada equation by Maple Symbolic packages are obtained. By employing Hirota’s bilinear technique, the extended soliton solutions according to bilinear frame equation are received. For this model, the contemplated model can be got by multi‐D binary Bell polynomials (bBPs). In addition, the analytical analysis of the high‐order soliton outcomes to present the discipline of outcomes. The effect of the free parameters on the behavior of acquired figures of a few obtained solutions for the nonlinear rational exact cases was also discussed. The above technique could also be employed to get exact solutions for other nonlinear models in physics, applied mathematics, and engineering.

Symmetry reduction, conservation laws and power series solution of time-fractional variable coefficient Caudrey–Dodd–Gibbon–Sawada–Kotera equation

Symmetry reduction, conservation laws and power series solution of time-fractional variable coefficient Caudrey–Dodd–Gibbon–Sawada–Kotera equation

ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

Employing Hirota’s bilinear form to find novel lump waves solutions to an important nonlinear model in fluid mechanics

In this paper, Hirota’s bilinear form has been employed to find novel lump waves solutions for the generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation. This equation is one of the most widely used equations in the field of fluid mechanics. Using the employed technique in the paper, several different categories of solutions to the equation are retrieved. Although these solutions have distinct structures, but all of them have emerged under the banner of the same method. This feature is one of the advantages of the method compared to other methods. 3D diagrams of some of the resulting solutions have also been added to the article. The techniques can be easily adopted in solving other partial differential equations.

New travelling wave analytic and residual power series solutions of conformable Caudrey–Dodd–Gibbon–Sawada–Kotera equation

In this research article, a novel analytical and numerical solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGDK) equation are established by means of Tanh method and 4th order residual power series method (RPSM). Dynamics and shapes of the extracted solutions are shown by constructing 3D surface plots. To better understand the mechanism of the CDGDK equation, 3D surface plots are developed to view the physical illustration of the extracted solutions with the selection of suitable values of the parameter. Comparisons of analytical and approximate solutions indicate that RPSM is very efficacious and reliable approach for exploring nonlinear models appearing in different areas of science and engineering.

Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns

Abstract In this research, we explore the dynamics of Caudrey–Dodd–Gibbon–Sawada–Kotera equations in (1 + 1)-dimension, such as N-soliton, and breather solutions. First, a logarithmic variable transform based on the Hirota bilinear method is defined, and then one, two, three and N-soliton solutions are constructed. A breather solution to the equation is also retrieved via N-soliton solutions. All the solutions that have been obtained are novel and plugged into the equation to guarantee their existence. 2-D, 3-D, contour plot and density plot are also presented.

Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

<abstract> This paper deals with localized waves in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation in the incompressible fluid. Based on Hirota's bilinear method, N-soliton solutions related to CDGKS equation are constructed. Taking the special reduction, the exact expression of multiple localized wave solutions comprising lump soliton(s) are obtained by using the long wave limit method. 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 localized waves. </abstract>

Exact traveling wave solutions of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

<abstract> In this work, we test the intgrability of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation on the Painlevé test and construct new Wick-type and nob-Wick-type versions of exact traveling wave solutions of the stochastic Wick-type fractional CDGSK equation by employing the Hermit transformation, the conformable fractional derivative and the sub-equations method. Moreover, we obtain exact traveling wave solutions of the fractional Sawada-Kotera (SK) equation and the fractional Caudrey-Dodd-Gibbon (CDG) equation as well. It is note that physical illustration may be useful to predict internal structure of the considered equations. The results confirm that sub-equations method is very effective and efficient to find exact traveling wave solutions of Wick-type fractional nonlinear evolution equations. </abstract>

Characteristics of the interaction behavior between solitons in (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Starting with Hirota bilinear form of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation, a class of lump solutions, a strip soliton, a pair of resonance solitons as well as the rogue wave have been obtained through symbolic computation. Firstly, two lump solutions are derived, consisting of six free parameters, while the other three accord with non-zero determinant condition which guarantee locality and analyticity for the solutions. Secondly, the interaction phenomenon between a lump wave and a stripe soliton is reflected. Interaction figures show that the lump wave can be absorbed by the stripe wave or separate from it. Thirdly, by a novel ansatz of hyperbolic function and quadratic function, a fresh transformation for the rogue wave and a pair of resonance solitons have been discovered, that a peak profiled rogue wave emerges from one of the resonance wave and transfers to another, and finally fades away. At the same time, numerical simulations are demonstrated to visualize the mechanism of the above solutions.

M-lump and lump–kink solutions of $$(2+1)$$-dimensional Caudrey–Dodd-Gibbon–Kotera–Sawada equation

In this paper, M-lump and interaction between lumps and kink solitons of the $$(2+1)$$ -dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation are studied based on Hirota bilinear form. M-lump solutions are derived by taking a ‘long wave’ limit of the N-soliton solutions, and the lump–kink solutions are presented consequently. In addition, evolutions of solutions are shown by choosing certain parameters.

Analytical Solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation via Symbol Calculation Approach

In this paper, we derive analytical solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation via symbol calculation approach. Applying the exp−φz-expansion method, we achieve the trigonometric, exponential, hyperbolic, and rational function solutions for the CDGSK equation. By choosing the appropriate parameters, we give some computer simulation to the analytical solutions of the CDGSK equation.