Exact Solutions, Hamiltonian Structures And Painlevé Property For Nonlinear Evolution Equations

This dissertation studies some topics in soliton and integrable system including the exact solutions, construction of the finite-dimensional integrable Hamiltonian system and applications of the Painleve property.Chapter 2 is devoted to study of exact solutions of the nonlinear evolution equations. Using solutions of a Bernoulli equation instead of tanh(kz) in tanh-function method we find some more general solutions of the KdV-Burgers-Kuramoto equation , and by using the nonlinear telegraph equation we show that there are many different choices on its balancing number m and the power n of the nonlinear term in Bernoulli equation by which we can recover the previously known solutions and also can derive new square root type solitary wave solutions. Exact solitary wave solutions for a surface wave equation are obtained by means of the homogeneous balance method. We also present an approach for constructing the solitary wave solutions and non-solitary wave solutions of the nonlinear evolution equations by using the homogeneous balance method directly, which is also used to find the steady state solutions, solitary wave solutions and the non-solitary wave solutions of the 2+1 dimensional dispersive long wave equations. The soliton-like solutions of the BLMP equation and the 2+1 dimensional breaking soliton equation are found by use of the symbolic-computation-based Method. We proposed a direct method for finding the exact solutions of the sine-Gordon type equations, and the exact solitary wave solutions of the sine-Gordon equation, double sine-Gordon equation, sinh-Gordon equation, MKdV-sine-Gordon equation and the Born - Infeld equation are found by our new method.Chapter 3 studies the method for constructing the finite-dimensional integrable Hamiltonian systems. By introducing the so-called modified Jacobi-Ostrogradsky coordinates, the non-regular constrained flows of the AKNS hierarchy and the Dirac hierarchy are transformed into the finite-dimensional integrable Hamiltonian systems, and their r-matrices, classical Pois-son structures, the second set of integral of motion and the complete integrability are presented.Using the gauge transformation between AKNS spectral problem and the Geng spectral problem, the gauge equivalent relation of the constrained flows are described, and the Hamiltonian structures, Lax representation and r-matrices of the constrained flows of the Geng hierarchy are constructed from the AKNS hierarchy. Taking the classical Boussinesq hierarchy as an illustrative example we show that a constrained flow may have different Hamiltonian forms under different choices of the coordinate system.Chapter 4 investigates the Painleve property. The auto-Backlund transformation and the exact solutions of the compound KdV-Burgers equation and a class of diffusion equation are obtained by using the truncated Painleve expansion.