On the Two - Dimensional Nonlinear. Korteweg - de Vries Equation. with Cubic Stream Function

Transcription

1 Advanced Studies in Theoretical Physics Vol. 0, 06, no. 4, HIKARI Ltd, On the Two - Dimensional Nonlinear Korteweg - de Vries Equation with Cubic Stream Function Sergei Evgenievich Ivanov ITMO National Research University (ITMO University Department of Information Systems 970 Saint Petersburg, 49 Kronverksky Pr., Russian Federation Vitaly Gennadievich Melnikov ITMO National Research University (ITMO University Head of the Department of Theoretical and Applied Mechanics 970, Saint Petersburg, 49 Kronverksky Pr., Russian Federation Copyright 06 Sergei Evgenievich Ivanov and Vitaly Gennadievich Melnikov. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In the theory of nonlinear oscillations and waves with dissipation and dispersion of the medium, it is applied the two-dimensional Korteweg - de Vries equation. In this paper a generalized nonlinear partial differential equation of third order with a cubic stream function is considered. The exact analytical solutions of two-dimensional Korteweg - de Vries equation and Kadomtsev - Petviashvili equation are obtained. The graphs effect of non-linear parameters of medium to the maximum amplitude of the waves is presented. Keywords: two-dimensional nonlinear equation of Korteweg - de Vries, the cubic stream function, equation of Kadomtsev Petviashvili Introduction The two-dimensional differential equation of Korteweg - de Vries and equation

2 58 Sergei E. Ivanov, Vitaly G. Melnikov of Kadomtsev - Petviashvili represent mathematical model for the description of nonlinear wave processes with dispersion and dissipation. In the present paper a model of nonlinear weak stationary traveling wave in the viscous medium is considered. The nonlinear solitary waves known as solitons in the interaction keep their shape without breaking. In the investigation of nonlinear differential equations in partial derivatives [8-5] have applied various analytical and numerical methods: the method of averaging, the method of small parameter, the perturbation methods, the method of separation of variables, the linearization method, the method of the stretched coordinates, the method of composite expansions, grid methods - the method of finite differences and finite elements method [6-4]. For the classical Korteweg - de Vries equation analytic solution is constructed using the method of inverse problem of scattering. The articles [-7] investigate the cases where the traveling wave for the Korteweg - de Vries equation exists and it is stable. Unlike other works we consider the two-dimensional Korteweg - de Vries equation with a cubic function of stream. The differential equation is a mathematical model for the description of nonlinear phenomena in dissipative media with dispersion. Solution of nonlinear equation of Korteweg - de Vries The mathematical model is presented by two-dimensional nonlinear differential equation of Korteweg - de Vries. + h + = f x t x x x 3 u u Fu ( u 3 ( The unknown function utx (,, x is depends on three variables. The stream function is represented by the cubic form: F( u = au + bu du 3 By substituting the cubic form in the equation ( we write the equation: 3 3 u u u u + h + ( a + bu du f 3 = x t x x x ( We introduce a new variable: y= ωt+ kx + kx + k3 (3 The equation ( with new variable can be written as: 3 u 3 u u k u ω + hk + ( a + bu du k 3 = f y y y y k y (4 The solution we will find in the form of hyperbolic function: u = α + β tanh( y (5

3 On the two - dimensional nonlinear Korteweg - de Vries equation 59 We is substituted the form (5 into the equation (4 and perform differentiation. βsec h ( y ( tanh( y( ak + k β tanh( y( b dα + bk α dk α + 3 dk β tanh ( y fk kω hk sinh(3 ysech ( y sech ( ( β( α + tanh( ( β k y b d y hk d The equation (6 is converted to the form sinh( y(cosh( y ak + bk α dk α dk β fk + 4hk + k ω + ( ak + bk α dk α + 3dk β fk 0 hk + k ω + α β β = ( b d (3k cosh( y k cosh(3 y 0 From the transformed equation (7 we get the system for determining the unknown coefficients: αβω.,, ak + bk α dk α + 3dk β fk 0hk + kω ak + bk α dk α dk β fk + 4hk + kω b dα b> 0, d 0, h> 0, k 0, k 0 we obtain the coefficients: For the case ( 4 ( α = b d, β = ± k 3 hα b, ω = -ak + 4hk + fk -bk α k For the case ( b 0, d 0, h 0, k 0, k 0 = > > we obtain the coefficients: 4 ( α, β = ± k 6 h d, ω = ak + hk + fk k By substituting the coefficients in the form (5 we get in the original variables the exact solution for the two-dimensional Korteweg-de Vries equation. b 6hk t ( 4adk b k + 4dfk + 8dhk u, = ± tanh + kx + kx + k3 (9 d d 4dk The figure shows the graph of solution of the equation with following parameters: a =, b=, d =, f =, h =. (6 (7 (8 Fig. Solutions of Korteweg - de Vries equation

4 60 Sergei E. Ivanov, Vitaly G. Melnikov For parameters a =, b=, d =, f =, h = the solution is form: ( ( 4 u.5 ± 6k tanh t 8k 5k + 4k 4k + kx + kx + k 3 The figure shows graphs of the maximum amplitude of wave from the nonlinear parameters. With increasing parameter h and decreasing the parameter d the amplitude of wave increases. Fig. The dependence of the maximum amplitude by parameters The graphs show that the amplitude is tends to zero and the wave is destroyed in cases of the following values: a = 4, f = From the solution (9 are obtained the ratio of the parameters for which the wave is destroyed. a = b k t + 4dk 4tdk tanh b (k 6dh + ( ( ( 4 ( 4dfkt + 8dhkt + 4dkx+ 4dkkx + 4dk3k ( 4dkt k b 3 f = 4adk t + b kt 4d tanh 8dhkt 4dkx 4dkx 4dk3 4dkt k 6dh To determine the constants k, k, k 3 in the solution (9, we use the conditions the initial boundary problem: 3 3 u u u(0,0,0,,, t > 0, L > 0, L > 0, x ( 0, L, x ( 0, L, x x (0,0,0 (0,0,0

5 On the two - dimensional nonlinear Korteweg - de Vries equation 6 By substituting conditions in the form of the solution (9 we obtain the system for determine the constants: b + 6 d hktanh( k k tanh ( k3sech ( k3 k sech ( k k tanh ( k3sech ( k3 k sech ( k3 The particular case of two-dimensional Korteweg - de Vries equation is equation of Kadomtsev - Petviashvili. The equation of Kadomtsev - Petviashvili can be written as: 3 u u 6 u u 3γ u 3 = (0 x t x x x The properties of the equation (0 depend on the dispersion of medium, which is determined by the parameterγ. The solution we will find in the form of a hyperbolic function: u(t, x, x = α + β tanh kx + k x + k + tω = α + β t anh ( y, ( ( 0 where y= kx + kx + k0+ tω By substitution the form ( in the equation (0 we find: 6 k ( 3α + 9β ( α + 5 βcosh( y + ( α + βcosh(4 y + γ ω = k cosh ( y(cosh( y 4k cosh ( y(cosh( y + 4 k ( 6cosh( y cosh(4 y 33 0 Let us transform equation ( to the exponential form: 8αk + 54βk 9γ k + 3kω+ 3k + ( 4y 4y ( e e ( 3αk 3βk 3 γ k / kω/ k y y ( e + e ( αk βk γ k + kω k = ( (3 From the equation (3 we obtain a system for determining the unknown parameters: 8αk + 54βk 9γ k + 3kω+ 3k 3αk + 3βk + 3 γ k / kω/ + k (4 6αk 30βk 3γ k + kω 5k From the system (4 we defined the required parameters α = ( 3γ k 4 kω 8k 6 k, β = k By substituting the parameters in the form of ( we obtain an exact solution for the equation of Kadomtsev Petviashvili:

6 6 Sergei E. Ivanov, Vitaly G. Melnikov 4 ( γ ω ( ω ut (, x, x = 3 k + k + 8k 6k k tanh kx + kx + k + t, (5 0 where k constants are determined from the initial and boundary conditions The form (5 shows the quadratic dependence of the solution on the parameter of the dispersion -γ. Figure 3 shows the graph of the solution of Kadomtsev - Petviashvili with parameter γ = 3 The graph shows the form of the nonlinear traveling wave. Conclusion Fig.3 Solutions of Kadomtsev - Petviashvili equation The two-dimensional nonlinear differential equation of the Korteweg-de Vries with a cubic function of stream is considered. The generalized equation is the mathematical model for the description of wave processes in a medium with dissipation and dispersion in the theory of nonlinear oscillations and waves. The model is describes the nonlinear stationary weak traveling waves in the viscous environment. In this article we obtained the exact analytic solution and graphs of influence the nonlinear parameters of medium to the amplitude of waves. Acknowledgements. The research was carried out with the financial support of the Russian Foundation for Basic Research A. References [] M.S. Islam, K. Khan, M.A. Akbar, An analytical method for finding exact solutions of modified Korteweg-de Vries equation, Results in Physics, 5 (05, [] B. Xue, F. Li, G. Yang, Explicit solutions and conservation laws of the coupled modified Korteweg-de Vries equation, Physica Scripta, 90 (05,

7 On the two - dimensional nonlinear Korteweg - de Vries equation 63 no. 8, [3] N. Kim, Formal initial value problem of the Korteweg-de Vries equation, Journal of Physics A: Mathematical and Theoretical, 48 (05, no. 5, [4] V. Kotlyarov, A. Minakov, Modulated elliptic wave and asymptotic solitons in a shock problem to the modified Korteweg-de Vries equation, Journal of Physics A: Mathematical and Theoretical, 48 (05, no. 30, [5] L. Ma, H. Li, J. Ma, Single-peak solitary wave solutions for the generalized Korteweg de Vries equation, Nonlinear Dynamics, 79 (05, [6] F. Achleitner, C.M. Cuesta, S. Hittmeir, Travelling waves for a non-local Korteweg-de Vries-Burgers equation, Journal of Differential Equations, 57 (04, no. 3, [7] T. Dlotko, M.B. Kania, S. Ma, Korteweg-de Vries-Burgers system in R^N, Journal of Mathematical Analysis and Applications, 4 (04, no., [8] E. P. Kolpak, S. E. Ivanov, Mathematical modeling of the system of drilling rig, Contemporary Engineering Sciences, 8 (05, no. 6, [9] E. P. Kolpak, L.S. Maltseva, S. E. Ivanov, On the stability of compressed plate, Contemporary Engineering Sciences, 8 (05, no. 0, [0] I. V. Zhukova, E. P. Kolpak, Y. E. Balykina, Mathematical model of growing tumor, Applied Mathematical Sciences, 8 (04, [] S. A. Kabrits, E. P. Kolpak, Numerical Study of Convergence of Nonlinear Models of the Theory of Shells with Thickness Decrease, AIP Conference Proceedings, 648 (05, [] E. P. Kolpak, S. A. Kabrits, V. Bubalo, The follicle function and thyroid gland cancer, Biology and Medicine, 7 (05, BM [3] E.A. Kosjakov, A.A. Tikhonov, Differential equations for librational motion of gravity-oriented rigid body, Int. Journal of Non-Linear Mechanics, 73 (05,

8 64 Sergei E. Ivanov, Vitaly G. Melnikov [4] Y. G. Pronina, Study of possible void nucleation and growth in solids in the framework of the Davis-Nadai deformation theory, Mechanics of Solids, 49 (04, [5] E. P. Kolpak, S. E. Ivanov, Mathematical and computer modeling vibration protection system with damper, Applied Mathematical Sciences, 9 (05, [6] V.G. Melnikov, Chebyshev economization in transformations of nonlinear systems with polynomial structure, International Conference on Systems - Proceedings, (00, [7] V.G. Melnikov, N.A. Dudarenko, Method of forbidden regions in the dynamic system matrices root clustering problem, Automation and Remote Control, 75 (04, [8] S. E. Ivanov, V. G. Melnikov, Mathematical modeling vibration protection system for the motor of the boat, Applied Mathematical Sciences, 9 (05, no. 9, [9] C.B. Dolicanin, A.A. Tikhonov, On dynamical equations in s-parameters for rigid body attitude motion, Int. Conf. on Mech.-Seventh Polyakhov's Reading (05, [0] S. E. Ivanov, V. G. Melnikov, On the equation of fourth order with quadratic nonlinearity, International Journal of Mathematical Analysis, 9 (05, no. 54, [] G. I. Melnikov, S. E. Ivanov, V. G. Melnikov, The modified Poincare-Dulac method in analysis of autooscillations of nonlinear mechanical systems, Journal of Physics: Conference Series, 570 (04, [] E. P. Kolpak, L.S. Maltseva, Rubberlike membranes at inner pressure, Contemporary Engineering Sciences, 8 (05, no. 36, [3] B. Melnikov, A. Semenov, Fatigue damage accumulation under the complex varying loading, Applied Mechanics and Materials, 67 (04, [4] B.E. Melnikov, L.F. Hazieva, On the possibility of neglecting the coupled-mode oscillations of a single-mass dynamic system under non-

9 On the two - dimensional nonlinear Korteweg - de Vries equation 65 steady kinematic excitations, Advanced Materials Research, (04, Received: January 6, 06; Published: April 9, 06