Caudrey-Dodd-Gibbon-Sawada-Kotera Research Articles

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.

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.

A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation

Bernoulli sub‐equation function method is applied to obtain exact solutions of Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation. As a result of this, exact traveling‐wave and some new oscillating solutions to CDGSK are obtained. It may be observed that Bernoulli sub‐equation function method employed here is very effective and reliable to get explicit solutions for this nonlinear partial differential equation. Profiles of all constructed solutions are graphically illustrated entirely as well.

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.

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.

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>

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>

Novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method

In this work, the analytic solutions for different types of nonlinear partial differential equations are obtained using the multiple Exp-function method. We consider the stated method for the (3+1)-dimensional generalized shallow water-like (SWL) equation, the (3+1)-dimensional Boiti–Leon- Manna–Pempinelli (BLMP) equation, (3+1)-dimensional generalized variable-coefficient B-type Kadomtsev–Petviashvili (VC B-type KP) equation and the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation. We obtain multi classes of solutions containing one-soliton, two-soliton, and triple-soliton solutions. All the computations have been performed using the software package Maple. The obtained solutions include three classes of soliton wave solutions in terms of one-wave, two-waves, and three-waves solutions. Then the multiple soliton solutions are presented with more arbitrary autocephalous parameters, in which the one, two, and triple solutions localized in all directions in space. Moreover, the obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. The different types of obtained solutions of aforementioned nonlinear equations arising in fluid dynamics and nonlinear phenomena.

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.

New soliton solutions of conformable time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation in modeling wave phenomena

In this paper, new traveling wave solutions of time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation have been revealed by the help of extended Kudryashov’s method. In this case, the time fractional derivative has been considered in the sense of conformable fractional derivative. The salient feature of the proposed extended Kudryashov’s method is to take full advantage of solutions for the Riccati and the Bernoulli equations involving parameters. Using some particular wave transformation, the given equation is converted to the nonlinear ODE by the help of chain rule and the derivative of a composite function; and then, a novel analytical method viz. extended Kudryashov’s method has been used to solve the resulting equation. Consequently, some new exact solutions have been successfully obtained. The exact solutions devised by the proposed method manifest that the proposed method is effective and easy to implement.

Interaction behavior between solitons and (2+1)-dimensional CDGKS waves

Starting from the Darboux transformation (DT) related nonlocal symmetry of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation, the original symmetry is localized by introducing four field quantities. On the basis of the local symmetry, some novel group invariant solutions are derived by utilizing the classical symmetry reduction method. The result shows that the essential and unique role of the DT is to add an additional soliton to the fifth order nonlinear wave, which is the basic reduction of the (2+1)-dimensional CDGKS equation. As an illustration, the interactions between solitons and fifth order nonlinear waves expressed by Jacobi elliptic functions and incomplete elliptic integrals of the third kind are exhibited, and some of their dynamical properties are analyzed.

Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation

In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov’s nonlocal conservation method is applied to construct Cls for time fractional CDGDK.

Interaction Behaviours Between Solitons and Cnoidal Periodic Waves for (2+1)-Dimensional Caudrey—Dodd—Gibbon—Kotera—Sawada Equation**Supported by the National Natural Science Foundation of China under Grant No. 11505154, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ16A010003, and the Scientific Research Foundation for Doctoral Program of Zhejiang Ocean University under Grant No. Q1511

The consistent tanh expansion (CTE) method is employed to the (2+1)-dimensional Caudrey—Dodd—Gibbon-Kotera—Sawada (CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.

Non-traveling wave solutions for the (2+1)-D Caudrey-Dodd-Gibbon-Kotera-Sawada equation

Purpose – The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation. Design/methodology/approach – The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation. Findings – The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable. Research limitations/implications – As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found. Originality/value – These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.

Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

The integrability of an extended fifth-order KdV equation with Riccati-type pseudopotential

The extended fifth-order KdV equation in fluids is investigated in this paper. Based on the concept of pseudopotential, a direct and unifying Riccati-type pseudopotential approach is employed to achieve Lax pair and singularity manifold equation of this equation. Moreover, this equation is classified into three categories: extended Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, extended Lax equation and extended Kaup–Kuperschmidt (KK) equation. The corresponding singularity manifold equations and auto-Backlund transformations of these three equations are also obtained. Furthermore, the infinitely many conservation laws of the extended Lax equation are found using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas.

ANALYTICAL INVESTIGATION OF THE CAUDREY–DODD–GIBBON–KOTERA–SAWADA EQUATION USING SYMBOLIC COMPUTATION

In this paper, the Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation is analytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its N-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Bäcklund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation.

A class of Neumann type systems and its application

A class of Neumann type systems are derived separating the spatial and temporal variables for the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation and the modified Korteweg-de Vries (mKdV) hierarchy. The Lax-Moser matrix of Neumann type systems is worked out, which generates a sequence of integrals of motion and a hyperelliptic curve of KdV type. We deduce the constrained Hamiltonians to put Neumann type systems into canonical Hamiltonian equations and further complete the Liouville integrability for the Neumann type systems. We also specify the relationship between Neumann type systems and infinite dimensional integrable systems (IDISs), where the involutivity solutions of Neumann type systems yield the finite parametric solutions of IDISs. From the Abel-Jacobi variables, the evolution behavior of Neumann type flows are shown on the Jacobian of a Riemann surface. Finally, the Neumann type flows are applied to produce some explicit solutions expressed by Riemann theta functions for the 2+1 dimensional CDGKS equation and the mKdV hierarchy.

Finite-gap solutions of 2+1 dimensional integrable nonlinear evolution equations generated by the Neumann systems

Each soliton equation in the Korteweg–de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN−1 of the unit sphere SN−1. The Lax–Moser matrix for the Neumann systems of degree N−1 is deduced in view of the Mckean–Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax–Moser matrix, the constrained Hamiltonians in the sense of Dirac–Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN−1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN−1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.