Characterizations of all real solutions for the KdV equation and $W_{\mathbb{R}}$

Highlights

• Characterizations of all real functions in class W are given.

• Real traveling wave solutions of the KdV equation are characterized.

• Develop a complex analytic method to express real traveling wave solutions of NLDEs.

Abstract

With the development of the study on nonlinear differential equations (NLDEs), more and more researchers consider exact meromorphic solutions of NLDEs in class $W$, which consists of elliptic function or their degeneration. It is meaningful to study the real-valued solutions of NLDEs. In this paper, we give characterizations of all real solutions for NLDEs in class $W$. As an example, we apply it to the KdV equation to show the sharpness of our results.

Introduction

Nonlinear partial and ordinary differential equations describe lots of physical scenarios which occur in many areas of plasma physics, nonlinear optics, fluid dynamics and engineering. It is important to study exact solutions of nonlinear differential equations so as to better understand the phenomena modeled by the underlying NLDEs. In order to construct exact solutions to NLDEs, a number of algorithms and methods have been developed, such as the inverse scattering method [1], [2], Hirota bilinear method [3], [4], dynamical systems method [5], similarity transformation method [6], three wave method [7], $(G^{\prime }∕G)$-expansion method [8], sine–Gordon expansion method [9], exp function method [10], etc.

It is difficult to obtain the solutions of differential equations in the complex plane because of the existence of singularities. Moreover, there are many types of singularities, such as pole, essential singularity, movable singularity, logarithmic singularity, etc. The behaviors of the solutions of NLDEs neighboring these singularities are also very complicated. Among these singularities, the properties of poles are among the best, so it is significant to study meromorphic solutions of NLDEs. Many researchers have studied meromorphic solutions of NLDEs and achieved a lot of meaningful results. For example, Eremenko et al. [11], [12] investigated the following $k$th-order Briot–Bouquet equation

$$F(w,w^{\left(k\right)})=\sum _{j=0}^m{P}_j\left(w\right){\left(w^{\left(k\right)}\right)}^j=0,$$

where $P_j\left(w\right)$ are polynomials of $w$ with constant coefficients. They proved that any meromorphic solution of Eq. (1) with at least one pole on $ℂ$ belongs to the class $W$ whether $k$ is odd [11] or even [12]. So far, meromorphic solutions having at least one pole of Briot–Bouquet equation are clear. If a meromorphic function $g$ is a rational function of $z$, or a rational function of  $e^{\mu z},\mu \in ℂ$, or an elliptic function, then $g$ is said to belong to the class $W$. Combining the local singularity analysis and global estimation on the meromorphic solutions, Eremenko [13] proved that all meromorphic solutions of the traveling wave reduction of the Kuramoto–Sivashinsky equation belong to the class $W$. Eremenko gave the classification method of the meromorphic solutions of NLDEs, and provided the idea of constructing meromorphic exact solutions. Following their work, Musette and Conte [14] initially proposed the subequation method to construct meromorphic solutions of NLDEs. Demina and Kudryashov [15] introduced a powerful method, combining Mittag-Leffler theorem and theory of elliptic functions, to seek meromorphic exact solutions of some NLDEs. After these methods are established, they are applied to construct meromorphic exact solutions of many partial differential equations, such as the Kawahara equation [16], real cubic Swift–Hohenberg equation [17], complex cubic–quintic Ginzburg–Landau equation [18], variant Boussinesq equations [19], Jimbo–Miwa equation [20], etc. However, in general it is not easy to show whether a given ordinary differential equation (ODE) satisfies the conditions proposed by Eremenko [13] especially for higher-order ODEs. More recently, Ng and Wu [21] showed that there exists a special class of autonomous algebraic ODEs consisting of any order satisfy these conditions and hence all their meromorphic solutions belong to the class $W$. Most of these algorithms consider the solutions on the complex plane. Nevertheless, it is very useful in physics and interesting in mathematics to study the real-valued solutions of NLDEs. Although their work enriches and further improves the methods of finding exact solutions of NLDEs, we still need to know under what circumstances the solution found is real. Therefore an important problem is to classify families of real solutions.

Whether from the perspective of complex analysis or from the perspective of the applications in physics and other fields, it is of great significance to characterize the real-valued solutions of nonlinear differential equations, which is the aim of this paper.