Higher-Order Equations of the KdV Type are Integrable

Abstract

We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent with an integrable equation which has recently appeared in the literature. We also discuss the integrability of both second- and third-order approximations of additional cases.

1. Introduction

The one-dimensional motion of solitary waves of inviscid and incompressible fluids has been the subject of research for more than a century [1]. Probably, one of the most important results in the above study was the derivation of the famous Korteweg-de Vries (KdV) equation [2]

𝑢_𝑡+𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=0. (1.1) At first, the equation was difficult to be examined due to the nonlinearity. The first important step was made with the numerical discovery of soliton solutions by Zabusky and Kruskal [3]. Soon thereafter great progress was made by the discovery of the Lax Pair representation [4] and the Inverse Scattering Transform [5]. The laters results led to a new notion of integrability. More specifically, according to this notion, a partial differential equation is said to be “completely integrable” if it is linearizable through a Lax Pair; thus, it is solvable via the Inverse Scattering Transform (see, e.g., [6]).

Equation (1.1) represents a first-order approximation in the study of long wavelength, small amplitude waves of inviscid and incompressible fluids. Allowing the appearance of higher-order terms in 𝛼 and 𝛽, one can obtain more complicated equations. Two such equations, including second- and third-order terms, were proposed in [7, 8] and have, respectively, the forms

𝑢_𝑡+𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥+𝛼²𝜌₁𝑢²𝑢_𝑥+𝛼𝛽(𝜌₂𝑢𝑢_𝑥𝑥𝑥+𝜌₃𝑢_𝑥𝑢_𝑥𝑥𝑢)=0,_𝑡+𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥+𝛼²𝜌₁𝑢²𝑢_𝑥+𝛼𝛽(𝜌₂𝑢𝑢_𝑥𝑥𝑥+𝜌₃𝑢_𝑥𝑢_𝑥𝑥)+𝛼³𝜌₄𝑢³𝑢_𝑥+𝛼²𝛽(𝜌₅𝑢²𝑢_𝑥𝑥𝑥+𝜌₆𝑢𝑢_𝑥𝑢_𝑥𝑥+𝜌₇𝑢³_𝑥)=0. (1.2)(1.3)

Equation (1.2) was first examined both analytically and numerically in [9]. The violation of the Painlevé property in many cases, together with a numerical study of the reduction 𝑢=𝑢(𝑥) in the complex 𝑥-plane, gave strong indications that, in general, this equation is not integrable. Consequently, (1.2) was examined in [10, 11] and it was found that it possesses solitary wave solutions, which, for small values of the parameters 𝛼 and 𝛽, behave like solitons. New wave solutions of both (1.2) and (1.3) were also examined numerically in [12] and were also found to behave like solitons.

Equation (1.2) was further examined in [13–20], while (1.3) was examined in [18, 21, 22]. Although an enormous amount of new solutions was presented, no progress has been made regarding the integrability of these equations.

In this paper we show that, for arbitrary 𝜌₁ and

𝜌₂=4𝜌₁,𝜌₃=2𝜌₁,𝜌₄=0,𝜌₅=𝜌₆=4𝜌²₁,𝜌₇=−8𝜌²₁9, (1.4) equation (1.3) is equivalent to an integrable equation recently proposed by Qiao and Liu [23]. We, thus, reveal an integrable case of (1.3) itself. We also discuss the existence of additional integrable cases for both (1.2) and (1.3).

2. An Integrable Case of (1.3)

Recently, Qiao and Liu [23] proposed a new integrable equation, namely,

𝑚_𝑡=12(1𝑚²)_𝑥𝑥𝑥−12(1𝑚²)_𝑥. (2.1) The integrability follows directly from the fact that the equation admits a Lax Pair; thus, as mentioned above, is solvable by the Inverse Scattering Transform.

It is quite easy to prove that (2.1) is actually a subcase of (1.3). More specifically, we first set

𝑚=𝑣^−2/3, (2.2) thus, (2.1) becomes

𝑣_𝑡−𝑣²𝑣_𝑥+𝑣²𝑣_𝑥𝑥𝑥+𝑣𝑣_𝑥𝑣_𝑥𝑥−29𝑣³_𝑥=0. (2.3) We then set

𝑢=𝑎₁𝑣(𝑋,𝑡)+𝑎₂,𝑋=𝑎₃𝑥+𝑎₄𝑡, (2.4) where 𝑎_𝑖 are constants; substitute in (1.3), divide by 𝑎₁, and write 𝑥 instead of 𝑋. We thus obtain

𝑣_𝑡+[𝑎₃(1+𝛼𝑎₂+𝛼²𝑎²₂𝜌₁+𝛼³𝑎³₂𝜌₄)+𝑎₄]𝑣_𝑥+𝛼𝑎₁𝑎₃(1+2𝛼𝑎₂𝜌₁+3𝛼²𝑎²₂𝜌₄)𝑣𝑣_𝑥+𝛽𝑎³₃(1+𝛼𝑎₂𝜌₂+𝛼²𝑎²₂𝜌₅)𝑣_𝑥𝑥𝑥+𝛼²𝑎²₁𝑎₃(𝜌₁+3𝛼𝑎₂𝜌₄)𝑣²𝑣_𝑥+𝛼𝛽𝑎₁𝑎³₃[(𝜌₂+2𝛼𝑎₂𝜌₅)𝑣𝑣_𝑥𝑥𝑥+(𝜌₃+𝛼𝑎₂𝜌₆)𝑣_𝑥𝑣_𝑥𝑥]+𝛼³𝑎³₁𝑎₃𝜌₄𝑣³𝑣_𝑥+𝛼²𝛽𝑎²₁𝑎³₃(𝜌₅𝑣²𝑣_𝑥𝑥𝑥+𝜌₆𝑣𝑣_𝑥𝑣_𝑥𝑥+𝜌₇𝑣³_𝑥)=0. (2.5) Clearly, (2.3) and (2.5) are equivalent if the following terms vanish:

𝐴₁=𝑎₃(1+𝛼𝑎₂+𝛼²𝑎²₂𝜌₁+𝛼³𝑎³₂𝜌₄)+𝑎₄,𝐴₂=𝛼𝑎₁𝑎₃(1+2𝛼𝑎₂𝜌₁+3𝛼²𝑎²₂𝜌₄𝐴),₃=𝛽𝑎³₃(1+𝛼𝑎₂𝜌₂+𝛼²𝑎²₂𝜌₅𝐴),₄=𝛼²𝑎²₁𝑎₃(𝜌₁+3𝛼𝑎₂𝜌₄𝐴)+1,₅=𝛼𝛽𝑎₁𝑎³₃(𝜌₂+2𝛼𝑎₂𝜌₅𝐴),₆=𝛼𝛽𝑎₁𝑎³₃(𝜌₃+𝛼𝑎₂𝜌₆𝐴),₇=𝛼³𝑎³₁𝑎₃𝜌₄,𝐴₈=𝛼²𝛽𝑎²₁𝑎³₃𝜌₅𝐴−1,₉=𝛼²𝛽𝑎²₁𝑎³₃𝜌₆𝐴−1,₁₀=𝛼²𝛽𝑎²₁𝑎³₃𝜌₇+29. (2.6)

Since 𝛼𝛽𝑎₁𝑎₃≠0, relations 𝐴₇=0, 𝐴₂=0, and 𝐴₆=0 imply, respectively, that

𝜌₄=0,𝑎₂1=−2𝛼𝜌₁,𝜌₆=2𝜌₁𝜌₃, (2.7) while relations 𝐴₃=0 and 𝐴₅=0 imply that

𝜌₂=4𝜌₁,𝜌₅=4𝜌²₁. (2.8) Then, relations 𝐴₄=0, 𝐴₁=0, 𝐴₉=0, and 𝐴₁₀=0 yield, respectively,

𝑎₃1=−𝛼²𝑎²₁𝜌₁,𝑎₄=4𝜌₁−14𝛼²𝑎²₁𝜌²₁,𝜌₃𝛼=−⁴𝑎⁴₁𝜌²₁2𝛽,𝜌₇=2𝛼⁴𝑎⁴₁𝜌³₁.9𝛽 (2.9) Finally,

𝐴₈=0⇒𝑎⁴₁=−4𝛽𝛼⁴𝜌₁. (2.10) We, thus, conclude with relations (1.4), while 𝜌₁ remains arbitrary.

3. Discussion

In [9] it was shown that (1.2) does not pass the classical Painlevé test [24, 25] for any combination of the 𝜌_𝑖 parameters, except of course for the cases 𝜌₁=𝜌₂=𝜌₃=0 and 𝜌₂=𝜌₃=0 in which it reduces to KdV and modified KdV, respectively. However, as also stated in [9], there are infinitely many cases for which the equation has only algebraic singularities, that is, it admits the so-called weak-Painlevé property [26]. Although this property does not constitute a strong indication for integrability, there are still many integrable equations admitting only algebraic singularities (see, e.g., [27]).

On the other hand, at least to our knowledge, no results regarding integrable cases of (1.3) have appeared in the bibliography before. Equation (1.3) is highly nonlinear and most probably it is not integrable in general. However, as shown in the previous section, there is at least one nontrivial combination of the 𝜌_𝑖 parameters, for which it is completely integrable.

Based on the above statements, we believe that it would be interesting to study whether there are any integrable cases for (1.2) or any additional integrable cases for (1.3). We hope to present results in this direction in a future publication.