Fifth-order KdV Equation Research Articles

Symmetry Group Analysis of a Fifth-Order KdV Equation with Variable Coefficients

ABSTRACTSymmetry group analysis is carried out on a generalized fifth-order KdV (foKdV) equation involving many arbitrary functions. Equivalence transformations group has been determined. This allows us to perform a comprehensive study by reducing the equation to a subclass with fewer number of arbitrary elements. Furthermore, we have established the subclasses of the reduced equation which are nonlinearly self-adjoint. The property of nonlinearly self-adjointness is used to construct conserved vectors from the classical symmetries of the equation by using a general theorem on conservation laws. We also determine conservation laws by using the multipliers method.

The Integrability of an Extended Fifth-Order KdV Equation in 2+1 Dimensions: Painlevé Property, Lax Pair, Conservation Laws, and Soliton Interactions

Abstract In this article, we apply the singularity structure analysis to test an extended 2+1-dimensional fifth-order KdV equation for integrability. It is proven that the generalized equation passes the Painlevé test for integrability only in three distinct cases. Two of those cases are in agreement with the known results, and a new integrable equation is first given. Then, for the new integrable equation, we employ the Bell polynomial method to construct its bilinear forms, bilinear Bäcklund transformation, Lax pair, and infinite conversation laws systematically. The N-soliton solutions of this new integrable equation are derived, and the propagations and collisions of multiple solitons are shown by graphs.

Local well-posedness for the fifth-order KdV equations on [formula omitted

This paper is a continuation of the paper Low regularity Cauchy problem for the fifth-order modified KdV equations onT[7]. In this paper, we consider the fifth-order equation in the Korteweg–de Vries (KdV) hierarchy as following:{∂tu−∂x5u−30u2∂xu+20∂xu∂x2u+10u∂x3u=0,(t,x)∈R×T,u(0,x)=u0(x)∈Hs(T).We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev initial data via the energy method. This paper follows almost same idea and argument as in [7]. Precisely, we use some conservation laws of the KdV Hamiltonians to observe the direction which the nonlinear solution evolves to. Besides, it is essential to use the short time Xs,b spaces to control the nonlinear terms due to high × low ⇒ high interaction component in the non-resonant nonlinear term. We also use the localized version of the modified energy in order to obtain the energy estimate.As an immediate result from a conservation law in the scaling sub-critical problem, we have the global well-posedness result in the energy space H2.

Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping

A fifth-order KdV equation with time-dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations; for instance, they can be used to determine conservation laws. We obtain the symmetries of the model applying Lie’s classical method. The choice of some arbitrary functions of the equation by the equivalence transformation enhances the study of Lie symmetries of the equation. We have determined the subclasses of the equation which are nonlinearly self-adjoint. This allow us to obtain conservation laws by using a theorem proved by Ibragimov which is based on the concept of adjoint equation for nonlinear differential equations.

Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids

Abstract Korteweg–de Vries (KdV)-type equations are used as approximate models governing weakly nonlinear long waves in fluids, where the first-order nonlinear and dispersive terms are retained and in balance. The retained second-order terms can result in the extended fifth-order KdV equation. Through the Darboux transformation (DT), multi-soliton solutions for the extended fifth-order KdV equation with coefficient constraints are constructed. Soliton propagation properties and interactions are studied: except for the velocity, the amplitude and width of the soliton are not influenced by the coefficient of the original equation; the amplitude, velocity, and wave shape of each soltion remain unchanged after the interaction. By virtue of the generalised DT and Taylor expansion of the solutions for the corresponding Lax pair, the first- and second-order rational solutions of the equation are obtained.

Soliton collisions and integrable aspects of the fifth-order Korteweg-de Vries equation for shallow water with surface tension

The fifth-order Korteweg-de Vries (KdV) equation works as a model for the shallow water waves with surface tension. Through symbolic computation, binary Bell-polynomial approach and auxiliary independent variable, the bilinear forms, N -soliton solutions, two different Bell-polynomial-type Backlund transformations, Lax pair and infinite conservation laws are obtained. Characteristic-line method is applied to discuss the effects of linear wave speed c as well as length scales τ and γ on the soliton amplitudes and velocities. Increase of τ , c and γ can lead to the increase of the soliton velocity. Soliton amplitude increases with the increase of τ . The larger-amplitude soliton is seen to move faster and then to overtake the smaller one. After the collision, the solitons keep their original shapes and velocities invariant except for the phase shift. Graphic analysis on the two and three-soliton solutions indicates that the overtaking collisions between/among the solitons are all elastic.

The Interactions ofN-Soliton Solutions for the Generalized (2+1)-Dimensional Variable-Coefficient Fifth-Order KdV Equation

A generalized (2+1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1+1)-dimensional KdV equation. TheN-soliton solutions of the (2+1)-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficienteAij; wheneAij=0, the soliton fusion and fission will happen; wheneAij≠0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method.

Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method

Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.

Some remarks on the fifth-order KdV equations

In this paper, we present the differential operators for the generalized fifth-order KdV equation. We give formal proofs on the Hamiltonian properties including the skew-adjointness and Jacobi identity by using the prolongation method. Our results show that there are three third-order Hamiltonian operators which can be used to construct the Hamiltonians. However, no fifth-order operators are shown to pass the Hamiltonian test, although there are an infinite number of them, and they are skew-adjoint.

Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth-order KdV equation (0.1) ∂ t u − ∂ x 5 u = c 1 ∂ x u ∂ x 2 u + c 2 ∂ x ( u ∂ x 2 u ) + c 3 ∂ x ( u 3 ) , \begin{equation*} \tag {0.1} \partial _tu-\partial ^5_x u=c_1\partial _xu\partial _x^2u+c_2\partial _x(u\partial _x^2u)+c_3\partial _x(u^3), \end{equation*} where x ∈ R x \in \mathbb R , t ∈ R t \in \mathbb R , u = u ( x , t ) u=u(x,t) is a real-valued function and α , c 1 , c 2 , c 3 \alpha , \ c_1, \ c_2, \ c_3 are real constants with α ≠ 0 \alpha \neq 0 , is locally well-posed in H s ( R ) H^s(\mathbb R) for s ≥ 2 s \ge 2 . In the Hamiltonian case (i.e. when c 1 = c 2 c_1=c_2 ), the IVP associated to (0.1) is then globally well-posed in the energy space H 2 ( R ) H^2(\mathbb R) .

Group analysis of variable coefficient generalized fifth-order KdV equations

We carry out group analysis of a class of generalized fifth-order Korteweg-de Vries equations with time dependent coefficients. Admissible transformations, Lie symmetries and similarity reductions of equations from the class are classified exhaustively. A criterion of reducibility of variable coefficient fifth-order KdV equations to their constant coefficient counterparts is derived. Some exact solutions are presented.

Exact solutions for fifth-order KdV-type equations with time-dependent coefficients using the Kudryashov method

The KdV equation plays an important role in describing motions of long waves in shallow water under gravity, one-dimensional nonlinear lattice, fluid mechanics, quantum mechanics, plasma physics, nonlinear optics and other areas. The KdV equation is a well-known model for the description of nonlinear long internal waves in a fluid stratified by both density and current. The aim of this paper is to present solitary wave solutions of the fifth-order KdV equations with time-dependent coefficients. The Kudryashov method is applied to solve the governing equations and then exact 1-soliton solutions are obtained. It is shown that this method provides us with a powerful mathematical tool for solving high-order nonlinear partial differential equations with time-dependent coefficients in mathematical physics.

Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided.

Stability Analysis for Travelling Wave Solutions of the Olver and Fifth-Order KdV Equations

The Olver equation is governing a unidirectional model for describing long and small amplitude waves in shallow water waves. The solitary wave solutions of the Olver and fifth-order KdV equations can be obtained by using extended tanh and sech-tanh methods. The present results are describing the generation and evolution of such waves, their interactions, and their stability. Moreover, the methods can be applied to a wide class of nonlinear evolution equations. All solutions are exact and stable and have applications in physics.

Bäcklund Transformation and Quasi-Periodic Solutions for a Variable-Coefficient Integrable Equation

Binary Bell polynomials are applied to construct bilinear formalism, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of the generalized variable-coefficient fifth-order Korteweg-de Vries equation. In the meantime, quasi-periodic wave solutions for the equation are obtained by using the Riemann theta function. The asymptotic properties of one-periodic wave solution and two-periodic wave solutions are also established, respectively.

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.

Frobenius integrable decompositions of nonlinear evolution equations with modified term

Frobenius integrable decompositions are introduced for nonlinear evolution equations with modified term. Two classes of polynomial equations with modified term are transformed into Frobenius integrable ordinary differential equations by two special Bäcklund transformations. The resulting solutions are illustrated to describe the solution phenomena shared with the mKdV equation, Sharma–Tasso–Olver equation, the modified Kawachara equation, the fifth-order KdV equation and its various forms.

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.

Integrability for the generalised variable-coefficient fifth-order Korteweg–de Vries equation with Bell polynomials

In the present paper, a direct and unifying Bell polynomial approach is employed to find the bilinear representations, bilinear Bäcklund transformation and Lax pair of a generalised variable-coefficient fifth-order Korteweg–de Vries equation. Note that the integrable constraint conditions on the variable coefficients can be naturally found in the procedure of applying the Bell polynomials. In addition, with the help of the Hirota bilinear method, the N -soliton solutions of this equation are also obtained.

Rough solutions of the fifth-order KdV equations

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg–de Vries (KdV) hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,x,t∈R,u(0,x)=u0(x),u0∈Hs(R). We prove a priori bound of solutions for Hs(R) with s⩾54 and the local well-posedness for s⩾2. The method is a short time Xs,b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,b structures to reduce the contribution of high–low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we obtain that the fifth-order equation in the KdV hierarchy,∂tu−∂x5u−30u2∂xu+20∂xu∂x2u+10u∂x3u=0 is globally well-posed in the energy space H2.