Nonlinear Self-Adjointness for some Generalized KdV Equations
The new concepts of self-adjoint equations formulated in Gandarias (J Phys A: Math Theor 44:262001, 2011) and Ibragimov (J Phys A: Math Theor 44:432002, 2011) are applied to some classes of third order equations. Then, from Ibragimov’s theorem on conserva
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Nonlinear Self-Adjointness for some Generalized KdV Equations M.L. Gandarias and M. Rosa
Abstract The new concepts of self-adjoint equations formulated in Gandarias (J Phys A: Math Theor 44:262001, 2011) and Ibragimov (J Phys A: Math Theor 44:432002, 2011) are applied to some classes of third order equations. Then, from Ibragimov’s theorem on conservation laws, conservation laws for two generalized equations of KdV type and a potential Burgers equation are established. Keywords Self-adjointness • Conservation laws • Lie symmetries
1.1 Introduction The classical KdV equation arises in various physical contexts and it models weakly nonlinear unidirectional long waves. A more complicated equation is obtained if one allows the appearance of higher-order terms. This equation is non-integrable but still admits some special wave solutions [16]. This equation, ut C kux C ˛uux C ˇuxxx C ˛ 2 1 u2 ux C ˛ˇ.2 uuxxx C 3 ux uxx / D 0
(1.1)
which will be referred to as a generalized KdV equation, was studied in [3] by Fokas, who presented a local transformation connecting it with an integrable partial differential equation (PDE). The higher-order wave equations of KdV type model strongly nonlinear long wavelength and short amplitude waves. It is for the reason that the strongly nonlinear character and integrability of these equations attract many researchers to study them. In [19], for some special sets of parameters, the authors derived some analytical expressions for solitary wave solutions and they carried
M.L. Gandarias () • M. Rosa Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain e-mail: [email protected]; [email protected] J.A.T. Machado et al. (eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity 6, DOI 10.1007/978-3-319-01411-1__1, © Springer International Publishing Switzerland 2014
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M.L. Gandarias and M. Rosa
out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme. The integral bifurcation method was used in [15] to study (1.1) and some new travelling wave solutions with singular or nonsingular character were obtained for some special sets of parameters. In [16], Marinakis considered as well the third order approximation ut C kux C ˛uux C ˇuxxx C ˛ 2 1 u2 ux C ˛ˇ.2 uuxxx C 3 ux uxx / C ˛ 3 4 u3 ux C ˛ 2 ˇ.5 u2 uxxx C 6 uux uxx C 7 u3x / D 0:
(1.2)
Equation (1.2) is equivalent to an integrable equation recently studied in [17] and the study in [16] reveals two integrable cases for (1.2). After some changes of variables for particular values of the parameters, (1.2) is transformed into 4 ut C u2 ux C u3x uux uxx C u2 uxxx D 0 9
(1.3)
Recently Marinakis proved that (1.3) is integrable. In [6] (see also [5]), a general theorem on conservation laws for arbitrary differential equations which do not require the existence of Lagrangians has been proved. This new theorem is based on the concept of adjoint equations for nonlinear equations. Th
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