Finite depth effects on solitary waves in a floating ice sheet

A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff׳s hypothesis, which yields a conservative and nonlinear expression for the bending force. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases

Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation

The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7)

(R1992) RBF-PS Method for Eventual Periodicity of Generalized Kawahara Equation

In engineering and mathematical physics, nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations evidenced by findings link belonging to their long-term actions named as eventual time periodicity discovered over solutions to IBVPs (initial-boundary-value problems). Here we investigate the solution’s eventual periodicity for generalized fifth order Kawahara equation (IBVP) on bounded domain in combination with periodic boundary conditions numerically exploiting mesh-free technique called as Radial basis function pseudo spectral (RBF-PS) method

Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term

Nonlinear interaction of long-wave distrubances with short-scale oscillations in stratified flows

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1996.Includes bibliographical references (leaves 145-150).by Tian-Shiang Yang.Ph.D

Solving Nonlinear Partial Differential Equations by the sn-ns Method

We present the application of the sn-ns method to solve nonlinear partial differential equations. We show that the well-known tanh-coth method is a particular case of the sn-ns method