Fifth-order KdV Equation Research Articles

Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves

We study solitary wave solutions of the fifth-order Korteweg–de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived, and the dependence of its amplitude, width, and speed on the parameters of the governing equation is studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).

The solitary waves, quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation

ABSTRACTUnder investigation in this paper is a fifth-order Korteweg-de Vries (fKdV) equation, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the binary Bell polynomials, a lucid and systematic approach is proposed to systematically study its bilinear representation, bilinear Bäcklund transformations and Lax pairs with explicit formulas, respectively. These results can be reduced to the ones of several integrable equations such as Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Lax equation, Kaup-Kuperschmidt equation and Ito equation, etc. Furthermore, the N-solitary wave solutions formula and quasi-periodic wave solutions are obtained by using bilinear form of the fKdV equation. Finally, the relation between the periodic wave solution and solitary wave solution is rigorously established.

Exact solutions via equivalence transformations of variable-coefficient fifth-order KdV equations

In this paper, a family of variable-coefficient fifth-order KdV equations has been considered. By using an infinitesimal method based on the determination of the equivalence group, differential invariants and invariant equations are obtained. Invariants provide an alternative way to find equations from the family which may be equivalent to a specific subclass of the same family and the invertible transformation which maps both equivalent equations. Here, differential invariants are applied to obtain exact solutions.

Exact solutions for a fifth-order two-mode KdV equation with variable coefficients

Exact solutions for a fifth-order two-mode KdV equation with variable coefficients

Quasi-periodic solution of quasi-linear fifth-order KdV equation

We prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth order KdV equation on the torus in presence of a quasi-periodic forcing.

The study of lump solution and interaction phenomenon to $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional generalized fifth-order KdV equation

In this paper, based on the Hirota bilinear method, a kind of lump solutions and two classes of interaction solutions are discussed to the $$(2+1)$$ -dimensional generalized KdV equation with the aid of symbolic computation system Mathematica. Analyticity is naturally guaranteed by taking special choices of the involved parameters to achieve a positive constant term. Particularly, these solutions with special values of the included parameters are plotted, as illustrative example.

Conservation Laws by using the Multiplier Method for a Fifth-Order Kdv Equation with Time-Dependent Coefficients and Linear Damping

In this paper we consider a family of fifth-order Korteweg-de Vries equations with time-dependent coefficients and linear damping term. By using the multiplier method of Anco and Bluman we determine all the low order conservation laws.

Symmetry Reductions for a Generalized Fifth Order KdV Equation

Abstract In this work, Lie symmetry analysis is performed on a generalized fifth-order KdV equation. This equation describes many nonlinear problems with great physical interest in mathematical physics, nonlinear dynamics and plasma physics, among them it is a useful model for the description of wave phenomena in plasma and solid state and internal solitary waves in shallow waters. Group invariant solutions are obtained which allow us to transform the equation into ordinary differential equations. Furthermore, taking into account the conservation laws that the ordinary differential equation admits we reduce the order of the equations. Finally, we obtain some exact solutions.

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny–Lin equation

This paper investigates the invariance properties of the time fractional Benny–Lin equation with Riemann–Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto–Sivashinsky equation and Navier–Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny–Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

Two New Energy-Preserving Algorithms for Generalized Fifth-Order KdV Equation

AbstractIn this paper, based on the multi-symplectic formulations of the generalized fifth-order KdV equation and the averaged vector field method, two new energy-preserving methods are proposed, including a new local energy-preserving algorithm which is independent of the boundary conditions and a new global energy-preserving method. We prove that the proposed methods preserve the energy conservation laws exactly. Numerical experiments are carried out, which demonstrate that the numerical methods proposed in the paper preserve energy well.

Improving Variational Iteration Method with Auxiliary Parameter for Nonlinear Time-Fractional Partial Differential Equations

In this research, we present a new approach based on variational iteration method for solving nonlinear time-fractional partial differential equations in large domains. The convergence of the method is shown with the aid of Banach fixed point theorem. The maximum error bound is specified. The optimal value of auxiliary parameter is obtained by use of residual error function. The fractional derivatives are taken in the Caputo sense. Numerical examples that involve the time-fractional Burgers equation, the time-fractional fifth-order Korteweg–de Vries equation and the time-fractional Fornberg–Whitham equation are examined to show the appropriate properties of the method. The results reveal that a new approach is very effective and convenient.

Exact solutions to the time-fractional differential equations via local fractional derivatives

This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa–Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.

Nonlocal Symmetries and Finite Transformations of the Fifth-Order KdV Equation

Abstract The n th finite transformations of the fifth-order KdV equation are obtained from the Lie point symmetry approach via localisation of nonlocal symmetries to local ones of the enlarged system. Through the obtained transformations, some periodic and soliton solutions are derived.

Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalizedexp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.

Modulating functions-based method for parameters and source estimation in one-dimensional partial differential equations

In this paper, modulating functions-based method is proposed for estimating space–time-dependent unknowns in one-dimensional partial differential equations. The proposed method simplifies the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of the modulating functions-based solution is proved. The wave and the fifth-order KdV equations are used as examples to show the effectiveness of the proposed method in both noise-free and noisy cases.

Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist

In this work we establish two wave modes for the integrable fifth-order Korteweg-de Vries (TfKdV) equations. We determine necessary conditions of the nonlinearity and dispersion parameters of the equation for multiple-soliton solutions to exist. We apply the simplified Hirota method to derive multiple-soliton solutions under these conditions. We also examine the dispersion relations and the phase shifts of the developed models.

Solitons, periodic waves, breathers and integrability for a nonisospectral and variable-coefficient fifth-order Korteweg–de Vries equation in fluids

Under investigation in this paper is a nonisospectral and variable-coefficient fifth-order Korteweg–de Vries equation in fluids. By virtue of the Bell polynomials and symbolic computation, the bilinear form, Bäcklund transformation and Lax pair are obtained. Based on its bilinear form, N -soliton solutions are constructed. Furthermore, periodic wave and breather wave solutions are obtained by virtue of the Riemann theta function and homoclinic test approach, respectively. In addition, the characteristic-line method is applied to discuss the features of the solitons for the nonisospectral problem, such as the amplitude, velocity and width of the solitary wave.

Soliton fusion and fission in a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids

Under investigation in this paper is a generalized variable-coefficient fifth-order Korteweg-de Vries equation, which describes the interaction between a water wave and a floating ice cover or the gravity-capillary waves. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, N-soliton solutions, Bäcklund transformation and Lax pair are derived. Infinitely-many conservation laws are obtained based on the Bell-polynomial-typed Bäcklund transformation. Soliton fusion and fission, and influence of the variable coefficients from the equation are analyzed: Both variable coefficients c(t) and n(t) are in direct proportion to the soliton velocities but have no effect on the amplitudes, while another constant coefficient α can affect the types of the interactions, in the sense of the elastic or inelastic. Elastic–inelastic interactions among the three solitons are presented as well.

A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions

A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions

Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain

In this paper, we consider the initial boundary value problem (IBVP) of the Hamiltonian fifth-order KdV equation posed on a finite interval $(0,L)$, $$ \begin{cases} \partial_t u-\partial_x^{5}u= c_1 u\partial_x u+ c_2 u^2 \partial _x u + 2b\partial_x u\partial_x^{2}u+bu\partial_x^{3} u, \quad x\in (0,L), \ t>0 \\ u(0,x)=\phi (x) , \ x\in (0,L)\\ u(t,0)=\partial_x u(t,0)=u(t,L)=\partial_x u(t,L)=\partial_x^{2}u(t,L)=0, \quad t>0, \end{cases} $$ and show that, given $0\leq s\leq 5$ and $T>0$, for any $\phi \in H^s (0,L) $ satisfying the natural compatibility conditions, the IBVP admits a unique solution $$ u\in L^{\infty}_{loc} (\mathbb R^+; H^s(0,L))\cap L^2 _{loc}(\mathbb R^+; H^{s+2} (0,L)). $$ Moreover, the corresponding solution map is shown to be locally Lipschtiz continuous from $L^2 (0,L)$ to $L^{\infty}(0,T; L^2 (0,L))\cap L^2 (0,T; H^2 (0,L))$ and from $H^5 (0,L)$ to $L^{\infty}(0,T; H^5 (0,L))\cap L^2 (0,T; H^7 (0,L))$, respectively, for any given $T>0$ . This is in sharp contrast to the pure initial value problem (IVP) of the equation posed on the whole line $\mathbb R$, $$ \begin{cases} \partial_t v-\partial_x^{5}v= c_1 v\partial_x v+ c_2 v^2 \partial _x v + 2b\partial_x v\partial_x^{2}u+bv\partial_x^{3} v, \quad x\in \mathbb R, \ t\in \mathbb R \\ v(0,x)=\psi (x) , \ x\in \mathbb R, \end{cases} $$ which is known to be (globally) well-posed in the space $H^s (\mathbb R)$ for $s\geq 2$ and the corresponding solution map is continuous, but fails to be uniformly continuous on any ball in $H^s(\mathbb R)$.