INTEGRABLE EVOLUTION EQUATIONS

Integrable evolution equations are certain nonlinear partial differential equations or semidiscrete nonlinear difference equations that are used to model wave propagation in various media. The goal of this thesis is to present the derivation of integrable evolution equations in a way accessible to nonexperts in the field of integrable systems and to illustrate those derivations by various explicit examples. In the case of nonlinear partial differential equations, both the spacial variable x and temporal variable t are continuous independent variables. In the case of semidiscrete nonlinear difference equations, the spacial variable n is a discrete independent variable and the temporal independent variable t is a continuous variable. In the continuous case, the spacial variable x takes all real values and the temporal variable t takes either nonnegative values or all real values. To derive integrable evolution equations in our thesis, we present the following four methods in each of the continuous and semidiscrete cases: (1) the Lax method, (2) the AKNS method, (3) the alternate Lax method, (4) the alternate AKNS method.