^{1, 2, 3}

^{2}

^{4}

^{1}

^{2}

^{3}

^{4}

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

It is well known that the study of nonlinear wave equations and their solutions are of great importance in many areas of physics. Travelling wave solution is an important type of solution for the nonlinear partial differential equation and many nonlinear partial differential equations have been found to have a variety of travelling wave solutions.

The well-known Korteweg-de-Vries equation

The Camassa-Holm equation

The Degasperis-Procesi equation

In 1993, Rosenau and Hyman [

To the best of our knowledge, the problems of the well-posedness and the uniformly continuity of (

For any

The paper is organized as follows. In Section

In this section we investigate the periodic travelling wave solutions of (

If

System (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

In this section we construct a family of smooth travelling wave solutions of suitably high frequency and provide a precise estimate of their periods.

Let

The periodic solution of (

The above discussion can be summarized as follows.

For any

The next proposition gives precise estimates for the period of the solution

The period

The half period can be expressed as

We write

Next we need some estimates in order to obtain upper bounds for these solutions. We start with

Suppose

With the help of (

Further, we proceed with

For any

For the first derivative we get

We close the section with an interpolation argument to obtain the estimates for noninteger values of the Sobolev index in the norm

Let

Define two sequences of travelling wave solutions

It is sufficient to estimate the

Finally, the behavior at time

This work is supported by the Foundation of Guangxi Key Lab of Trusted Software, The State Key Laboratory of Integrated Services Networks open grant 2012, Guangxi Key Lab of Wireless Wideband Communication & Signal Processing open grant 2012, Guangxi Natural Science Foundation (no. 2011GXNSFA018134, no. 2013GXNSFBB053005, and no. 2013GXNSFAA019006), and the National Natural Science Foundation of China (no. 11161013, no. 11261013, and no. 11361017). The author wish to thank the anonymous reviewers for their helpful comments and suggestions.