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]

ut+ux+αuux+βuxxx=0.
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

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=4ρ12,ρ7=-8ρ129,
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 (<xref ref-type="disp-formula" rid="EEq1.3">1.3</xref>)

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

mt=12(1m2)xxx-12(1m2)x.
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,
thus, (2.1) becomes

vt-v2vx+v2vxxx+vvxvxx-29vx3=0.
We then set

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

vt+[a3(1+αa2+α2a22ρ1+α3a23ρ4)+a4]vx+αa1a3(1+2αa2ρ1+3α2a22ρ4)vvx+βa33(1+αa2ρ2+α2a22ρ5)vxxx+α2a12a3(ρ1+3αa2ρ4)v2vx+αβa1a33[(ρ2+2αa2ρ5)vvxxx+(ρ3+αa2ρ6)vxvxx]+α3a13a3ρ4v3vx+α2βa12a33(ρ5v2vxxx+ρ6vvxvxx+ρ7vx3)=0.
Clearly, (2.3) and (2.5) are equivalent if the following terms vanish:

A8=0⇒a14=-4βα4ρ1.
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 ρi 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 ρi 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.

WhithamG. B.KortewegD. J.de VriesG.On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary WavesZabuskyN. J.KruskalM. D.Interaction of “solitons” in a collisionless plasma and the recurrence of initial statesLaxP. D.Integrals of nonlinear equations of evolution and solitary wavesGardnerC. S.GreeneJ. M.KruskalM. D.MiuraR. M.Method for solving the Korteweg—de Vries equationAblowitzM. J.SegurH.FokasA. S.On a class of physically important integrable equationsFokasA. S.LiuQ. M.Asymptotic integrability of water wavesMarinakisV.BountisT. C.DuncanD. B.EilbeckJ. C.On the integrability of a new class of water wave equationsProceedings of the Conference on Nonlinear Coherent Structures in Physics and BiologyJuly 1995Edinburgh, UKHeriot-Watt UniversityMarinakisV.BountisT. C.Special solutions of a new class of water wave equationsTzirtzilakisE.XenosM.MarinakisV.BountisT. C.Interactions and stability of solitary waves in shallow waterTzirtzilakisE.MarinakisV.ApokisC.BountisT.Soliton-like solutions of higher order wave equations of the Korteweg—de Vries typeKhuriS. A.Soliton and periodic solutions for higher order wave equations of KdV type. IHongW.-P.wphong@cu.ac.krDynamics of solitary-waves in the higher order Korteweg—de Vries equation type (I)LiJ.RuiW.LongY.HeB.Travelling wave solutions for higher-order wave equations of KdV type. IIILongY.LiJ.RuiW.HeB.Traveling wave solutions for a second order wave equation of KdV typeMarinakisV.New solutions of a higher order wave equation of the KdV typeMarinakisV.vangelismarinakis@hotmail.comNew solitary wave solutions in higher-order wave equations of the Korteweg—de Vries typeLiJ.Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equationsRuiW.LongY.HeB.Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III)LiJ.WuJ.ZhuH.Traveling waves for an integrable higher order KdV type wave equationsLiJ.Dynamical understanding of loop soliton solution for several nonlinear wave equationsQiaoZ.LiuL.A new integrable equation with no smooth solitonsWeissJ.TaborM.CarnevaleG.The Painlevé property for partial differential equationsWeissJ.The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivativeRamaniA.DorizziB.GrammaticosB.Painlevé conjecture revisitedGilsonC.PickeringA.Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations