The Travelling-wave Solutions To Displacement Shallow Water Equations Based On The Hamilton System

The wave height can be up to twenty or thirty meters on the sea,which is the main load in coastal engineering,port,the safety of ships,oil platforms and so on.The theory and calculation method of nonlinear water wave get very great development during the past several decades.Choosing velocity or velocity potential as basic variable,many wave mathematical models has developed in Eulerian description based on perturbation expansion method.However,the describing method of these models is insufficient,especially when uneven bottoms and moving boundaries occur.Analytical and numerical methods are basic methods for solving water wave equations,and the exact travelling-wave solution got from analytical method can be used to describe the movement of wave profile well.However,due to complexity of the nonlinear equations,the explicit solutions can be deduced in very few cases.With the development of computer technology,numerical method is widely applied to nonlinear problems.However,a heap of discrete data we get often cover up the internal relations among variables.Thus seeking new theory and method comes to an important work.In this paper,the Hamilton variational principle is used to study the one dimensional nonlinear shallow water problem in Lagrange coordinate,which can describe the fluid morphology by tracing particle position.By using horizontal displacement as variable,mathematical expression of kinetic energy and potential energy is derived.Under the incompressible condition,the shallow water equation based on displacement is deduced by variational principle,and then straightforwardly solve it by using traveling wave transformation.Taking advantage of elliptic functions,the exact travelling-wave solutions of the constructed equation are analyzed,and the range of the solutions is expanded,while the relationship between the parameters and the solutions is discussed.Energy conservation is strictly met in this theoretical system which 'has the merit of symplectic conservation.Using proposed equation,various analytical travelling wave solutions are presented,including the cnoidal wave,the periodic sharp/anti-sharp wave,the loop wave,the sharp solitary wave,and other types of solution waves.Based on the Hamilton system and Lagrange coordinate,the displacement method breaks a new way for solving the hydrodynamic equation,and lays a foundation for the application of the symplectic system in fluid mechanics.