Order KdV Equation Research Articles

Elliptic solutions for higher order KdV equations

We study higher order KdV equations from the GL(2, ) ≅ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic N-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find N-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.

Approximate Solution of Space-Time Fractional KdV Equation and Coupled KdV Equations

The main goal of this article is to find the approximate solution of the space-time fractional order KdV (STFKdV) equation and Coupled KdV (STFCKdV) equations by using Homotopy analysis method (HAM...

Jacobi spectral discretization for nonlinear fractional generalized seventh-order KdV equations with convergence analysis

We developed a numerical scheme to solve the nonlinear fractional (NF) generalized seventh order KdV (sKdV) equation with time-space-fractional using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

Applications of dispersive analytical wave solutions of nonlinear seventh order Lax and Kaup-Kupershmidt dynamical wave equations

In this article, new travelling wave solutions are constructed for two applications of the seventh order KdV equations, the seventh order Lax-equation and the seventh order Kaup-Kupershmidt equation. Using the modified auxiliary equation of direct algebraic method, we study the considered nonlinear PDEs analytically. As a result, we found several stable and analytical solutions.

Strongly nonlinear perturbation theory for solitary waves and bions

Strongly nonlinear perturbation theory would seem to be an oxymoron, that is, a contradiction of terms. Nonetheless, we here describe perturbation methods for wave categories that are intrinsically nonlinear including solitons (solitary waves), bound states of solitons (bions) and spatially periodic traveling waves (cnoidal waves). Examples include the Kortweg-deVries and Benjamin-Ono equations with general power law nonlinearity and the Fifth Order KdV equation. The perturbation strategies include (ⅰ) the Gorshkov-Ostrovsky-Papko near-equal-amplitude soliton interaction theory (ⅱ) perturbation series in the Newton-homotopy parameter and (ⅲ) approximations for large values of the nonlinearity exponent. A long section places strongly nonlinear perturbation theory for waves in a larger context as a subset of unconventional perturbation expansions including phase transition theory in \begin{document}$ 4 - \epsilon $\end{document} dimensions, the \begin{document}$ \epsilon = 1/D $\end{document} expansion where \begin{document}$ D $\end{document} is the dimension in quantum chemistry, the renormalized quantum anharmonic oscillator, the Yakhot-Orszag expansion in the exponent of the energy spectrum in hydrodynamic turbulence, and the Newton homotopy expansion.

Conservation laws for a generalized seventh order KdV equation

In this paper, by applying the multiplier method we obtain a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending on seven arbitrary nonzero parameters. We apply the Lie method in order to classify all point symmetries admitted by the equation in terms of the arbitrary parameters. We find that there are no special cases of the parameters for which the equation admits extra symmetries, other than those that can be found by inspection (scaling symmetry and space and time translation symmetries). We consider the reduced ordinary differential equations and we determined all integrating factors of the reduced equation from the combined x - and t - translation symmetries. Finally, we observe that all integrating factors arise by reduction of the low-order multipliers of the generalized seventh-order KdV equation.

Forced capillary gravity surface waves over a bump – Critical surface tension case

This paper concerns forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with nonzero surface tension on the free surface and a small bump on a horizontal rigid flat bottom. It is known that if non-dimensional wave speed, called Froude number, is near 1 and a non-dimensional surface tension, called Bond number, is near 1/3, the KdV theory fails and a time dependent fifth order KdV equation, called the Kawahara equation, can be derived to study the wave motion on the free surface. In this paper both time independent and time dependent forms of the Kawahara equation with a forcing are studied numerically and theoretically and various numerical results are presented. References B. Buffoni. Infinitely many large amplitude homoclinic orbits for a class of autonomous hamiltonian systems. J. Differential Equations (121):109–120, 1995. doi:10.1006/jdeq.1995.1123 B. Buffoni and J. F. Toland. Global existence of homoclinic and periodic orbits for a class of autonomous hamiltonian systems. J. Differential Equations, (118):104–120, 1995. doi:10.1006/jdeq.1995.1068 A. V. Buryak and A. R. Champneys. On the stability of solitary wave solutions of the fifth-order kdv equation. Phys. Lett. A, (233):58–62, 1997. doi:10.1016/S0375-9601(97)00453-2 A. R. Champneys and J. F. Toland. Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous hamiltonian systems. Nonlinearity, (6):665–721, 1993. J. W. Choi, S. M. Sun, and M. C. Shen. Steady capillary-gravity waves on the interface of two-layer fluid over an obstruction-forced modified k-dv equation. J. Eng. Math., (28):193–210, 1994. doi:10.1007/BF00058436 J. W. Choi, S. M. Sun, and S. I. Whang. Steady surface waves on water over a bump with critical surface tension. accepted. J. W. Choi, S. M. Sun, and S. I. Whang. Supercritical surface gravity waves generated by a positive forcing. European J. Mech. B/Fluids, (27):750–770, 2008. doi:10.1016/j.euromechflu.2008.01.006 L. K. Forbes. Critical free surface flow over a semi-circular obstruction. J. Eng. Math., (22):3–13, 1988. doi:10.1007/BF00044362 J. Hunter and J. Scheurle. Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D, (32):253–268, 1988. doi:10.1016/0167-2789(88)90054-1 J.K Hunter and J.M. Vanden-Broeck. Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech., (134):205–219, 1983. doi:10.1017/S0022112083003316 T. Iguchi. A mathematical justification of the forced korteweg-de vries equation for capillary-gravity waves. Kyushu Journal of Mathematics, 60(2):267–303, 2006. doi:10.2206/kyushujm.60.267 J. W. Miles. Stationary, transcritical channel flows. J. Fluid Mech., (162):489–499, 1986. doi:10.1017/S0022112086002136 Y. Pomeau, A. Ramani, and B. Grammaticos. Structural stability of the korteweg-de vries solitons under a singular perturbation. Phys. D, (31):127–134, 1988. doi:10.1016/0167-2789(88)90018-8 J. J. Stoker. Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, Interscience Publishers, Inc., New York, 4 edition, 1957. S. M. Sun and M. C. Shen. Exponentially small estimate for a generalized solitary wave solution to the perturbed k-dv equation. Nonlinear Anal., (23):545–564, 1994. doi:10.1016/0362-546X(94)90093-0 L. N. Trefethen. Spectral Methods in MATLAB. SIAM, Philadelphia, PA, 2003. J. M. Vanden-Broeck. Free surface flow over a semi-circular obstruction in a channel. Phys. Fluids, (30):2315–2317, 1987.

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.

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.

Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves

Abstract In this work, we aim to apply a numerical approach based on Homotopy perturbation transform method (HPTM) for derive the exact and approximate solutions of nonlinear fifth order KdV equations for study magneto-acoustic waves in plasma. The approach is a mixed form of the standard Laplace transform with the classical Homotopy perturbation technique. Nonlinear term can be handled with the aid of He's Polynomials. In this technique, the solution is calculated in the form of a convergent series and convergence of the HPTM solutions to the exact solutions is shown. The Homotopy perturbation transform method presents a wide applicability to handling nonlinear wave equations in science and engineering. As a novel application of HPTM, the present work shows some essential difference with the existing similar method. Several examples are provided for illustrate the simplicity and reliability of the method and highlighted the significance of this work.

The Multi-Wave Method for Exact Solutions of Nonlinear Partial Differential Equations

In this research, we use the multi-wave method to obtain new exact solutions for generalized forms of 5th order KdV equation and fth order KdV (fKdV) equation with power law nonlinearity. Computations are performed with the help of the mathematics software Mathematica. Then, periodic wave solutions, bright soliton solutions and rational function solutions with free pa rameters are obtained by this approach. It is shown that this method is very useful and effective.

The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions

The higher order nonlinear Schrödinger (NLS) equation describes ultra-short pluse propagation in optical fibres. The generalized nonlinear fifth-order of KdV equations derived from the higher order NLS equation by using multiple scales methods. We obtained the traveling wave solutions for some different kinds of the generalized nonlinear fifth-order of KdV (fifth-order Lax; fifth-order Ito; fifth-order Sawada–Kotera; fifth-order Kaup–Kupershmidt; fifth-order Caudrey–Dodd–Gibbon) equations by applying the auxiliary equation of the direct algebraic method. These solutions for the generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves by making the graphs of the exact solutions are analyzed. All solutions are exact and stable.

Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa-Holm equation

In this paper, we propose a new approach to calculate multi-soliton solutions of Camassa-Holm (CH) equation and modified Camassa-Holm (MCH) equation with the aid of Darboux transformation (DT). The new approach simplifies the approach presented by Li and Zhang [Proc. R. Soc. London, Ser. A 460, 2617–2627 (2004)]. We first map the CH and MCH equation to a negative order KdV (NKdV) equation by a reciprocal transformation. Then we proceed to apply the DT to solve the NKdV equation in the usual way. Finally we invert the reciprocal transformation to recover the solutions of the CH equation and MCH equation.

Two-dimensional Legendre wavelet method for travelling wave solutions of time-fractional generalized seventh order KdV equation

In this paper, a new method based on the Legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time-fractional seventh order KdV (sKdV) equation. Two-dimensional Legendre wavelet method is applied to compute the numerical solution of nonlinear time-fractional sKdV equation. The approximate solutions of nonlinear time fractional sKdV equation thus obtained by Legendre wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of the sKdV equation.

Application of HAM to seventh order KdV equations

In this paper, the homotopy analysis method (HAM) is applied to obtain the approximate solutions of KdV equations of seventh order, which are Sawada Kotera Ito equation, Lax equation, and Kaup–Kuperschmidt equation, respectively. The convergence of the homotopy analysis method is discussed with the help of auxiliary parameter h, which controls the convergence of the method and also called as the convergence control parameter. The results obtained by the HAM are compared with the exact solutions, by fixing the value of arbitrary constant. It is found that HAM is very robust and elegant method, and by choosing a suitable value of h we can get the approximate solution in very few iteration. Computations are performed with the help of symbolic computation package MATHEMATICA.

Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation

We study the generalized fifth order KdV equation using group methods and conservation laws. All of the geometric vector fields of the special fifth order KdV equation are presented. By using the nonclassical Lie group method, it is show that this equation does not admit nonclassical type symmetries. Then, on the basis of the optimal system, the symmetry reductions and exact solutions to this equation are constructed. For some special cases, we obtain additional nontrivial conservation laws and scaling symmetries.

Homotopy Perturbation Transform Method with He’s Polynomial for Solution of Coupled Nonlinear Partial Differential Equations

AbstractIn this paper, a combined form of the Laplace transform method with the homotopy perturbation method (HPTM) is applied to solve nonlinear systems of partial differential equations viz. the system of third order KdV Equations and the systems of coupled Burgers’ equations in one- and two- dimensions. The nonlinear terms can be easily handled by the use of He’s polynomials. The results shows that the HPTM is very efficient, simple and avoids the round-off errors. Four test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM) which shows that this method is a suitable method for solving systems of partial differential equations.

Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation $$\begin{aligned} \partial _tu - \partial _x^5u -30u^2\partial _xu + 20\partial _xu\partial _x^2u + 10u\partial _x^3u = 0. \end{aligned}$$ The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.

New solutions from nonlocal symmetry of the generalized fifth order KdV equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11347183, 11405110, 11275129, and 11305106) and the Natural Science Foundation of Zhejiang Province of China (Grant Nos. Y7080455 and LQ13A050001).

The nonlocal symmetry of the generalized fifth order KdV equation (FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Bäcklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.