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Contents
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Week 1: |
Linear Differential Expressions and Boundary Conditions, Differential Operator, Lagranges Formula, Adjoint Differential Expressions and Adjoint Boundary Conditions, Adjoint Differential Operator |
Week 2: |
Eigenvalues, Eigenfonctions and Associated Functions of Differential Operator |
Week 3: |
Green Functians and Inversion of a Differential Operator |
Week 4: |
Boundary-value Problems involving a Parameter and their reduction to an Integral Equation |
Week 5: |
Normalizing of a Boundary Conditions, Regular Boundary Conditions |
Week 6: |
Asymptotic Behaviour of Eigenvalues and Eigenfonctions |
Week 7: |
Expansion in term of Eigenfonctions of a Self-Adjoint Differential Operator, Expansion in term of Eigenfonctions and Associated Functions of a Differential Operator with Regular Boundary Conditions |
Week 8: |
Midterm exam. The Solutions of Boundary-value Problems for Partial Differential Equations by Fouriers Method |
Week 9: |
Integrable Nonlinear Evolition Equations with (1+1) and (2+1) dimensions, Solitary Waves and Solitons |
Week 10: |
Lax Method for Characterization of Exact Solvability, Lax pair for Korteweg-de Vries (KdV)equation |
Week 11: |
AKNS Method for Characterization of Exact Solvability, Lax pair for Sine-Gordon and Nonlinear Schrodinger Equation |
Week 12: |
Scattering and Inverse Scattering Problems for One-Dimensional Schrodinger Equation on the line |
Week 13: |
Scattering Data, Gelfand-Levitan-Marchenko Equation, Evolution of Scattering Data |
Week 14: |
Scheme of Inverse Scattering Method, N-Soliton Solution of KdV Equation |
Week 15*: |
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Week 16*: |
Final exam |
Textbooks and materials: |
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Recommended readings: |
N. A. Naimark, Linear Differential Operators; E. A. Coddington and N. Levinson, Theory of differential equations; M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform; V. A. Marchenko, Sturm-Liouville operators and applications |
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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