The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces

Abstract

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.

Travelling Wave Solutions for Three Dimensional Incompressible MHD Equations

“…where x y z t ξ α β = + + + , and change the Equations (4)- (10) into the following ordinary differential equations . The MHD equations govern the dynamics of the velocity and the magnetic field in electrically-conducting fluids and reflect the basic physics laws of conservation.…”

“…Aside from purely academic pursuits, MHD Journal of Applied Mathematics and Physics also plays an important role in the development of engineering technologies. Designing suitable engineering systems using electrically conducting fluids requires using computational techniques.One of the most prominent reasons for this difficulty is the phenomenon of fluid turbulence which again rears its head in MHD[9] [10]. In addition to the velocity field displaying disordered behavior the external electromagnetic field quantities also display such behavior[11].…”

Conserved Quantities and Fluxes for Some Nonlinear Evolution Equations

“…This discovery created renewed interest in the equations for solitary waves and the special properties of their solutions. New more powerful methods for describing the waves mathematically have been developed, and many equations have been found to have solitary waves and solitons as solutions [11][12][13][14][15][16][17][18][19][20]. In the mid 1920"s, Oskar Klein and Felix Gordon [21][22][23] derived an equation for a charged particle in an electromagnetic field, using thennew ideas in the realm of quantum theory.…”

“…However, the KdV equation has infinitely many conserved quantities. The existence of an infinite sequence of conservation laws for a given system of partial differential equations (PDEs) suggests that it is completely integrable, though such a condition is not required [25]. Indeed, there are systems (such as the Burgers equation) that can be directly integrated, though possess only a finite number of conservation laws [26].…”

Exact Solitary Wave Solutions for the Generalized Schamel Korteweg–De Vries Equation

“…Thus, it is important to investigate the exact explicit solutions of NLEEs. In recent years, various powerful methods have been presented for finding exact solutions of the NLEEs in mathematical physics, such as modified simple equation method (Bhrawy, et al, 2013), extended F-expansion method (Ma, 1993), tanh-sech method (Malfliet, 1992;Khater, et al, 2002;Wazwaz, 2006), extended tanh method (Ma & Fuchssteiner, 1996;El-Wakil & Abdou, 2007;Fan, 2000;Maliet, 2004), sine-cosine method (Wazwaz, 2004 a;Wazwaz, 2004b;Yusufoglu & Bekir, 2006) and Bä cklund transformation (Ma & Lee, 2009;Khater, et al, 2006;Khater, et al, 2004;Sayed, 2013) but solving nonlinear equations is still an important task. Some of the nonlinear models in plasma and dust plasma are described by canonical models, such as the KdV, the mKdV, and so on (Hassan, 2010).…”