Integrability Of Several Classical And Super Systems

This thesis is devoted to the integrability of several classical and super systems.Firstly,using symmetry approach,we investigate the 2nd-order,3rd-order and 5th-order 0-homogeneous scalar evolutionary supersymmetric equations,and present all equations ad-mitting higher order generalized symmetries.Under supersymmetric reciprocal transformation and changes of dependent variables,those equations are converted to potential supersymmetric(SUSY)Modified Korteweg-de Vries(MKdV)equation,potential SUSY 3rd-order Burgers e-quation,potential SUSY 5th-order MKdV equation,potential SUSY Fordy-Gibbons equation,SUSY Kawamoto equation.In addition to these well-known equations,we obtain two new SUSY equations,one is SUSY 3rd-order Harry Dym equation,the other is SUSY 5th-order Harry Dym equation which is linked to potential SUSY 5th-order MKdV equation through supersymmetry reciprocal transformation.Secondly,conservation laws of the generalized Riemann equations are studied.Searching for conservation laws of nonlinear partial differential equations is an important topic in the study of mathematical physics.Two special classes of conserved densities involving arbitrary functions are constructed for the generalized Riemann equation at N=2.By considering certain reductions,the result is extended to the famous Gurevich-Zybin,Monge-Ampere,two-component Hunter-Saxton equations and supersymmetric Hunter-Saxton equation.Inspired by the case when N = 2,two special classes of conserved densities involving arbitrary functions are also explicitly formulated for the system at arbitrary N.Lastly,based on its known bi-Hamiltonian structures,by squared eigenfunction method we obtain a spectral problem for Kupershmidt's extended equation of long waves.Unlike common spectral problem,it does not depend on linearly the spectral parameter.It is shown that this spectral problem can be embedded into the energy dependent Schrodinger spectral problem.Furthermore,Miura transformations and modified systems are obtained.We apply the approach of tri-Hamiltonian duality to the Kupershmidt's extended equation of long waves and obtain two Camassa-Holm type systems.