Approximate Symmetry Reduction And Hyperbolic Functions Method Application

With progress and development of the modern science and technology, nonlinear science has gradually become a cross-discipline. Modern technology has played a pivotal role in providing a theoretical support to advance the technological innovation for industrial production. Physics and many other disciplines are now usually in the state of describing the issues in their field via the models of the nonlinear partial differential equations. Therefore, it becomes increasingly important to investigate the various nonlinear problems involved in these disciplines. We introduce the approximate reduction method and the hyperbolic function method to study approximate solutions and exact solutions of the given equations, respectively.The first chapter introduces the development of approximate symmetry method, which leads to the perturbed problems based on the perturbation theory. Combined with a brief introduction of the perturbation theory, weak perturbation symmetry reduction method is introduced. For the perturbed equations which do not contain a small parameter, we then introduce the approximate homotopy method. Moreover, we introduce the historical background of the hyperbolic function method.The second chapter describes the extended hyperbolic function method and the KP equations. We also introduce the concept of traveling wave solutions and the method of balancing the equation. Via the hyperbolic function method, we can further obtain the exact solutions of a variety of evolution equations.The third chapter introduces the K(n,l) equation which leads to the K(n,l) equation with damping. We rewrite the K(n,l) equation as a system of equations with damping via the linear homotopy model combined the law of perturbation theory. Then the symmetry and the direct method were applied to the transformed system to obtain its approximate solution. The correspondence of the results is also derived.