JID:PLA AID:21996 /SCO Doctopic: Nonlinear science [m5Gv1.5; v 1.97; Prn:27/05/2013; 10:16] P.1(1-7) Physics Letters A ••• (••••) •••–••• Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Cnoidal waves governed by the Kudryashov–Sinelshchikov equation Merle Randrüüt a,∗ , Manfred Braun b a Tallinn University of Technology, Faculty of Mechanical Engineering, Department of Mechatronics, Ehitajate tee 5, 19086 Tallinn, Estonia b University of Duisburg–Essen, Chair of Mechanics and Robotics, Lotharstraße 1, 47057 Duisburg, Germany article info abstract Article history: Received 27 February 2013 Received in revised form 14 May 2013 Accepted 19 May 2013 Available online xxxx Communicated by R. Wu Keywords: Evolution equation Kudryashov–Sinelshchikov equation Korteweg–de Vries equation Phase curves Solitary waves Cnoidal waves The evolution equation for waves propagating in a mixture of liquid and gas bubbles as proposed by Kudryashov and Sinelshchikov allows, in a special case, the propagation of solitary waves of the sech 2 type. It is shown that these waves represent the solitary limit separating two families of periodic waves. One of them consists of the same cnoidal waves that are solutions of the Korteweg–de Vries equation, while the other one does not have a corresponding counterpart. It is pointed out how the ordinary differential equations governing traveling-wave solutions of the Kudryashov–Sinelshchikov and the Korteweg–de Vries equations are related to each other.  2013 Elsevier B.V. All rights reserved. 1. Introduction The nonlinear, one-dimensional flow of a mixture of liquid and gas bubbles has been analyzed for the first time by van Wijngaar- den [1] based on his own earlier linear theory. According to this linear theory, waves of small amplitude satisfy the same disper- sion equation as long gravity waves on a fluid of finite depth. Van Wijngaarden attributes this observation to Brooke Benjamin, who also concluded that the generalization to waves of finite ampli- tude in a liquid with gas bubbles must then lead to an equation of the Korteweg–de Vries type, in analogy to the nonlinear theory of long water waves. This nonlinear theory has been elaborated by van Wijngaarden [1]. Since then the dynamic behavior of liquids with gas bubbles remained a research field of enduring interest, mainly due to the many potential applications ranging from chemical engineering to biomedical science. A brief overview of the literature can be found in the introduction of [2]. Quite recently Kudryashov and Sinelshchikov [2,3] have recon- sidered the dynamics of a mixture of liquid and gas bubbles taking into account viscosity and heat transfer. In the case of vanishing dissipation, the propagation of nonlinear one-dimensional waves is described by the evolution equation v τ + α vv ξ + v ξξξ = ( vv ξξ ) ξ + β v ξ v ξξ , (1) * Corresponding author. Tel.: +372 6203209. E-mail address: merler@cens.ioc.ee (M. Randrüüt). which has gained considerable attention meanwhile and is known as the Kudryashov–Sinelshchikov (KS) equation. The dependent variable v represents the relative change of the pressure in the mixture or, alternatively, the relative change of the bubble radius. The dimensionless constants α and β depend, in a rather com- plicated way, on the material properties of the fluid and the gas bubbles. Special solutions representing undistorted traveling waves have been provided already in the original papers [2,3] and were stud- ied in detail by Randrüüt [4]. One of these solutions, namely for the parameter β =−3, has the form of a sech 2 solitary wave as it is known from the Korteweg–de Vries (KdV) equation. Actually this solution is already included, as a special case, in Kudryashov and Sinelshchikov’s paper [2], although in hidden form. Their Eq. (5.14), after correcting the sign and setting β =−3, assumes the form y = 1 − 1 C 3  1 − tanh 2  √ −α 2 (z + C 2 )  = 1 − 1 C 3 sech 2  √ −α 2 (z + C 2 )  , z = ξ − ατ . (2) Also one of Ryabov’s solutions [5] coincides with it. His Eq. (3.8) can be written as y = 1 + a 1 e z (1 + e z ) 2 = 1 + a 1 4 sech 2 z 2 , z = √ −α(x − αt ). (3) In both cases, the original notations have been kept. As can be seen from the definitions of the auxiliary variables z the solu- tions represent solitary waves traveling at the speed c = α. To get 0375-9601/$ – see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.05.040