Negative-order KdV Equation Research Articles

On a Riemann–Hilbert problem for the negative-order KdV equation

In this paper, we study the initial value problem for the negative order integrable KdV (NKdV) equation by Riemann–Hilbert problem method. The solutions of the NKdV equation are constructed in terms of the solution of a 2×2-matric Riemann–Hilbert problem via the asymptotic behavior of the spectral variable at one singularity point, λ=∞. And the one-soliton and two-soliton solutions are discussed in detail.

Negative Order KdV Equation with No Solitary Traveling Waves

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.

The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations

The double (G^( ^' )/G,1/G)-expansion method is used to find exact travelling wave solutions to the fractional differantial equations in the sense of Jumarie’s modified Riemann- Liouville derivative. We exploit this method for the combined KdV- negative-order KdV equation (KdV-nKdV) and the Calogero-Bogoyavlinskii-Schiff equation (CBS) of fractional order. We see that these solutions are concise and easy to understand the physical phenomena of the nonlinear partial differential equations. These solutions can be shown in terms of trigonometric, hyperbolic and rational functions.

American Institute of Mathematical Sciences

This work aims to study the negative-order KdV equation in (3+1)-dimensions which is developed via using the recursion operator of the KdV equation by employing the three-wave methods. As a consequence, a variety of novel multiwave solutions with several arbitrary parameters to the considered equation are presented. Moreover, selecting particular values for the parameters, some graphs are plotted to show the spatial structures and dynamics of the resulting solutions. These results enrich the variety of the dynamics in the field of nonlinear waves.

A variety of negative-order integrable KdV equations of higher orders

ABSTRACTWe develop a variety of negative-order integrable KdV equations of higher orders. We use the inverse recursion operator to construct these new equations. The complete integrability of each established equation is investigated via the Painlevé test, where each equation shows distinct branch of resonances. We use the simplified form of the Hirota’s direct method to obtain multiple soliton solutions for the generalized negative-order KdV equation.

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.

The quasi-periodic solutions of mixed KdV equations

The quasi-periodic solutions of mixed KdV equations are constructed. To derive the solutions, we introduce the mixed finite-dimensional integrable Hamiltonian systems (FDIHS), which are linearized on the Jacobi variety of the associated hyperelliptic curve. A relation between the FDIHS and the mixed KdV equations is revealed. Based on this relation, we construct the quasi-periodic solutions for the mixed KdV equations with aid of the Riemann theta function. The quasi-periodic solutions for the KdV6 equation and the negative order KdV equation are obtained as special members of the mixed KdV equations.

The first integral method for constructing exact and explicit solutions to nonlinear evolution equations

Problems that are modeled by nonlinear evolution equations occur in many areas of applied sciences. In the present study, we deal with the negative order KdV equation and the generalized Zakharov system and derive some further results using the so‐called first integral method. By means of the established first integrals, some exact traveling wave solutions are obtained in a concise manner. Copyright © 2012 John Wiley & Sons, Ltd.