Symmetries And Applications Of Related Transform Theory

In this dissertation we mainly investigate symmetries and their algebraic structures for some nonlinear intcgrable systems with special features. Transform theory of symme-tries will be used in sonic steps. These systems and special features arc:·Generalized Manakov equation and Sasa-Satsuma equation:Both arc physically important optics models but related to 3rd-order spectral problems which are different from the usual.·KdV equation with time-dependent coefficients:This equation is Painleve inte-grable under some restriction to coefficients, under which there is a gauge transformation between the equation and the usual KdV equation.·Toda lattice equation:Two potentials go to (1,0) instead of (0.0) when |n|→∞.We start from the symmetries and algebras for the generalized Manakov equation and Sa.sa-Satsuma equation. Generalized Manakov equation is essentially a generalized inte-grablc coupled nonlinear Schrodinger system and Sasa-Satsuma equation is a third order equation. Both equations arc related to 3 x 3 AKNS spectral problem (with 4 potentials) and can be reduced out from the 3x3 AKNS hierarchy. We construct symmetries and their Lie algebraic structures for the two equations from those of the 4-potential AKNS hierar-chy by reduction. During reduction the number of potentials is reduced to 2 and 1 from 4, respectively. So the key point is to guarantee the closeness of algebraic structure. For this we make use of a transformation during the reduction for the Sasa-Satsuma equation. Besides, by impleetic-sympleetic factorization of recursion we discuss multi-Hamiltonian structures for multi-component (4-potential) AKNS hierarchies.Then we investigate symmetries for the variable coefficient KdV equation with Painleve integrable condition. We first derive isospectral and non-isospectral flows of the variable coefficient KdV hierarchy and derive N-soliton solution for a non-isospectral variable coef-ficient KdV equation via bilinear approach. Those flows are structured, in some sense, by a formal recursion operatorΦ. Then, making use of the hereditary property ofΦand the relation betweenΦand those flows, we construct two sets of symmetries for the isospeetral variable coefficient KdV hierarchy. Lie algebraic structure of symmetries is also obtained. All these results can be obtained by using gauge transform theory from those of usual KdV equation.Finally. for the Toda lattice we choose a suitable time evolution so that we can have non-isospectral flows which go to zero as|n|→∞. This is quite important for the zero-curvature representation theory. By this theory we get two sets of symmetries for each isospeetral equation in Toda lattice hierarchy, but these symmetries compose a Lie algebra with new structure.