Course Information

Course Name	Turkish	Nonlineer Dalgalar
	English	Nonlinear Waves
Course Code	MAT 468E	Credit	Lecture (hour/week)	Recitation (hour/week)	Laboratory (hour/week)
Semester	-	3	3	-	-
Course Language		English
Course Coordinator		Semra Ahmetolan
Course Objectives		The course objective is to give the information of nonlinear waves, the equations governing the propagation of these waves, especially Korteweg-de Vries (KdV) equation, and their solutions which play an important part in many branches of mathematics, in engineering and in other areas of science. In this connection, 1. To give knowledge and basic understanding of linear wave equations, dissipative waves and dispersion. 2. To introduce nonlinear wave equations and to give basic understanding of discontinuous solutions, 3. To teach the elementary solutions of KdV equation and the behavior of these solutions, 4. To introduce the scattering and inverse scattering problems to students. 5. To give an idea about the solitary wave and solitons by constructing the solutions of initial value problem for KdV equation, 6. To demonstrate the Lax formulation, Hirota’s method and Backlund transformation to students. 7. To introduce a brief knowledge about numerical methods to solve nonlinear evolution equations which admit solitary wave, soliton.
Course Description		Korteweg-de Vries i.e. KdV equation; linear wave equation, superposition of solutions. Linear dispersive wave equation, dispersion. The simplest nonlinear wave equation and discontinuous solutions. The balance between nonlinearity and dispersion, and the KdV equation. Elementary solutions of the KdV equation; the qualitative behaviours of the traveling wave solutions of the KdV equation. Description of solutions in terms of the Jacobian elliptic functions. Limiting behaviours of the cnoidal wave and the solitary wave solutions. The scattering and inverse scattering problems; the scattering problem, the inverse scattering problem, the solution of the Marchenko equation. The initial-value problem for the KdV equation. Construction of the solution, Solitary wave and two-soliton solutions. Further properties of the KdV equation;Conservation laws, Lax formulation and its KdV hierarchy, Hirota’s method, bilinear form of the KdV equation. Backlund transformations for the KdV equation. The Painleve property of the KdV equations and numerical methods for the soliton solutions
Course Outcomes		Students completing this course will be able to I. Determine linear and nonlinear wave equations, dissipative waves and dispersion relation, II. Find the solutions of nonlinear equations and comment on the behavior of the solutions of these equations. III. Investigate scattering and inverse scattering problems, IV. Solve the general initial value problem for KdV equation. V. Use the Lax formulation, Hirota’ method and Backlund transformation , VI. Use numerical methods to find the solutions of nonlinear evolution equations, VII. Study the soliton solutions of nonlinear equations, numerically.
Pre-requisite(s)		MAT331/MAT331E
Required Facilities
Other
Textbook		P.G.Drazin and R.S.Johnson (1989), Solitons: an Introduction, Cambridge University Press
Other References		1. G.B. Whitham (1974), Linear and Nonlinear Waves, John Wiley& Sons. 2. R.K.Dodd, J.C. Eilbeck, J.D.Gibbon, H.C. Morris (1982), Solitons and Nonlinear Wave Equations, Academic Press, London

Courses

Help

About