Approximate Long Wave Equations Research Articles

Symmetry reductions and exact solutions to the systems of nonlinear partial differential equations**This work was supported by the National Natural Science Foundation of China under Grant No. 11171041 and the high-level personnel foundation of Liaocheng University under grant No. 31805.

In this paper, the Lie symmetry analysis is performed on the four systems of nonlinear partial differential equations (PDEs). One is the system of nonlinear coupled PDEs in a real physical application. The other system describes the unsteady liquid and gas flows in long pipelines. The third is the approximate equations for long waves in shallow water. The fourth is the general classical dispersiveless long wave equations. All of the point symmetries of the systems are presented for the first time, then the symmetry reductions and exact solutions to the systems are investigated.

Justification of the linear long-wave approximation to viscous fluid flow down an inclined plane

The objective of this paper is to justify the linear long-wave approximation used in the derivation of approximate equations for long waves on the free surface of a two-dimensional viscous fluid flow down an inclined plane. To the first order of a small parameter, the approximate equation is a heat equation, which becomes ill-posed if a Reynolds number R R is greater than some critical value R c {R_c} . To overcome this difficulty we consider a higher-order approximate equation, which is well-posed even if R > R c R > {R_c} , and show that the solution of the higher-order equation is an approximation to the solution of the linearized Navier-Stokes equations. The justification is based upon a set of long-wave initial conditions, and the error bounds can also be expressed in terms of pointwise estimates.

On a Korteweg–de Vries equation with variable coefficients in cylindrical coordinates

An equation of the Korteweg–de Vries (KdV) type with variable coefficients in cylindrical coordinates for nonlinear wave propagation on water of variable depth is presented. The special case concerning water motion with cylindrical symmetry reduces to an equation of the KdV type obtained from approximate equations for long waves.

The transformation of a solitary wave over an uneven bottom

Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.