DOI 10.1140/epjp/i2016-16356-3 Regular Article Eur. Phys. J. Plus (2016) 131: 356 T HE EUROPEAN P HYSICAL JOURNAL PLUS Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation Turgut Ak 1, a , S. Battal Gazi Karakoc 2 , and Houria Triki 3 1 Faculty of Engineering, Department of Transportation Engineering, Yalova University, 77100 Yalova, Turkey 2 Faculty of Science and Art, Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300 Nevsehir, Turkey 3 Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Received: 23 May 2016 / Revised: 29 August 2016 Published online: 12 October 2016 – c  Societ` a Italiana di Fisica / Springer-Verlag 2016 Abstract. In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable. 1 Introduction Interest in travelling-wave solutions for nonlinear partial differential equations (NLPDEs) has grown rapidly in recent years because of their importance in the study of complex nonlinear phenomena arising in dynamical systems. Such nonlinear wave phenomena appear in various fields of sciences, particularly in fluid mechanics, solid state physics, plasma physics and nonlinear optics. A variety of powerful methods have been developed to find analytical and numerical solutions of NLPDEs of all kinds. Examples include the Petrov-Galerkin method [1], the collocation method [2], the subsidiary ordinary differential equation method [3–5], Hirota’s method [6], the solitary wave ansatz method [7,8], Exp-function method [9], and many others. The well-known Korteweg-de Vries (KdV) equation [10] U t + aUU x + bU xxx =0, (1) where U is a real-valued function and a and b are real constants, is the generic model for the study of weakly nonlinear long waves [11]. It arises in physical systems which involve a balance between nonlinearity and dispersion at leading- order [12]. For example, it describes surface waves of long wavelength and small amplitude on shallow water and internal waves in a shallow density-stratified fluid [12]. In 1988, Philip Rosenau [13] introduced the Rosenau equation of the form U t + κU x + cU xxxxt + d ( U 2 ) x =0, (2) to describe the dynamics of dense discrete systems. For further consideration of the nonlinear wave, Zuo [14] added the viscous term U xxx to the Rosenau equation (2) and proposed the so-called Rosenau-KdV equation. The author obtained some solitons and periodic wave solutions of a e-mail: akturgut@yahoo.com