Negative‐order modified KdV equations: multiple soliton and multiple singular soliton solutions

In this work, we develop the negative‐order modified Korteweg–de Vries (nMKdV) equation. By means of the recursion operator of the modified KdV equation, we derive negative order forms, one for the focusing branch and the other for the defocusing form. Using the Weiss–Tabor–Carnevale method and Kruskal's simplification, we prove the Painlevé integrability of the nMKdV equations. We derive multiple soliton solutions for the first form and multiple singular soliton solutions for the second form. Copyright © 2015 John Wiley & Sons, Ltd.     

Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations

In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.     

Painlevé analysis and exact solutions of the Korteweg–de Vries equation with a source

We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painlevé test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.     

Integrability aspects of some two-component KdV systems

Some two-component Korteweg–de Vries systems are studied by prolongation technique and Painlevé analysis. Especially, the two-component KdV system conjectured to be integrable by Foursov is proved to be both Lax integrable and P-integrable. Its conservation laws are investigated based on the obtained Lax pair. Furthermore, it is shown that the three two-component Korteweg–de Vries systems are identical under certain invertible linear transformations. Finally, the auto-Bäcklund transformation and some exact solutions for the two-component Korteweg–de Vries system are derived explicitly.     

New exact solutions of a nonlinear integrable equation

Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp-function method, and Kudryashov method.     

Multiwave solutions to the negative-order KdV equation in (3+1)-dimensions

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 study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Meromorphic solutions of an auxiliary ordinary differential equation using complex method

In this paper, we employ the complex method to obtain first all meromorphic solutions of an auxiliary ordinary differential equation and then find all meromorphic exact solutions of the classical Korteweg–de Vries equation, Boussinesq equation, ( 3 + 1)‐dimensional Jimbo–Miwa equation, and Benjamin–Bona–Mahony equation. Our results show that the method is more simple than other methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple soliton solutions and other exact solutions for a two‐mode KdV equation

In this work, we study the two‐mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Absence of Global Solutions of the Korteweg–de Vries Equation

This paper is devoted to the problem of existence of global solutions of the Korteweg–de Vries equation. For certain initial–boundary problems for the Korteweg–de Vries equation, we obtain necessary conditions of existence (in other words, sufficient conditions of nonexistence) of global solutions.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Computing solutions to a forced KdV equation

In this paper a special forced Korteweg–de Vries (KdV) equation is considered. This equation is established by recent studies as a simple mathematical model of describing the physics of a shallow layer of fluid subject to external forcing. It serves as an analytical model of tsunami generation by submarine landslides. The bilinear form for this equation is obtained with the aid of Hirota’s method. Some of its one-, two- and three-soliton as well as breather-type soliton solutions and other interesting solutions are derived.     

A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation

In this article, we use a multilevel quartic spline quasi-interpolation scheme to solve the one-dimensional nonlinear Korteweg–de Vries (KdV) equation which exhibits a large number of physical phenomena. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this article also show that our scheme is feasible and valid.     

High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.     

General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfvén waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.     

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.     

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.     

On completely integrable coupled Burgers and coupled Korteweg–de Vries systems

In this work, we study two completely integrable equations, namely, coupled Burgers and Korteweg–de Vries systems. The modified form of Hirota’s bilinear method, established by Hereman, is employed to formally derive multiple-soliton solutions and multiple-singular-soliton solutions for each system. Hirota’s bilinear method is reliable and effective and can also be applied to solve other types of higher-dimensional integrable and non-integrable systems.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.