Soliton Solution Of Certain Nonlinear Differential Equations

Solitons are solitary waves which are known to propagate for very large distances in nonlinear media. Nonlinear equations govern many aspects of our lives. For example the large amplitude waves which we see at every sea shore are Solitons. In this thesis we have examined Soliton solutions of certain nonlinear differential equations (Nonlinear Schrödinger equation (NLS), Toda Lattice, KdV, Sine Gordon equation ,Perturbed Sine-Gordon equation and Navier Stokes equation) and the role they play in various physical systems (such as nonlinear optical lattices, Polyacetylene and Josephson Junctions). To understand the properties and behavior of the nonlinear equations they have to be solved exactly. This has been done in our first paper using Z transform. AKNS and ZS have shown that the temporal and spatial evolution of nonlinear differential equations can be understood in terms of equivalent coupled linear equations defined in a Hilbert space. This was strangely analogous to Quantum Mechanics where also the temporal development is described via a rotation in the Hilbert space. Obviously some similarity between Quantum Mechanics and Nonlinear Differential equation.