All traveling wave exact solutions of two kinds of nonlinear evolution equations

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Abstract

In this article, we employ the complex method to obtain all meromorphic solutions of complex Korteweg–de Vries (KdV) equation and the modified Benjamin–Bona–Mahony (mBBM) equation at first, then find out all traveling wave exact solutions of the Eqs. (KdV) and (mBBM). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions of the Eqs. (KdV) and (mBBM) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w2r,2(z) and simply periodic solutions w2s,1(z) which are not only new but also not degenerated successively by the elliptic function solutions. We give some computer simulations to illustrate our main results.

Section snippets

Introduction and main result

Nonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of sciences, particularly in fluid mechanics, solid state physics, plasma physics and nonlinear opticsExact solutions of NLPDEs of mathematical physics have attracted significant interest in the literature Over the last 20 years, much work has been done on the construction of exact solitary wave solutions, periodic wave solutions and analytic solutions of

Preliminary lemmas and the complex method

In order to give our complex method and the proofs of Theorem 1 and Theorem 2, we need some lemmas and results.

Lemma 1

[6], [7] Let kN, then any meromorphic solution wW of k order Briot-Bouquet equationsF(w(k),w)=i=0nPi(w)(w(k))i=0,where Pi(w) are polynomials with constant coefficients and w has at least one pole.

Set mN{1,2,3,},rjN0=N{0} r=(r0,r1,,rm) j=0,1,,m. DefineMr[w](z)[w(z)]r0[w(z)]r1[w(z)]r2[w(m)(z)]rm.p(r)r0+r1++rm is called the degree of Mr[w]. Differential polynomial P(w,w,,

Proofs of the main theorems

Proof of Theorem 1. Substituting (6) into the Eq. (1) we have q=2.p=1,c-2=-12βk2,c-1=0,c0=λ,c1=0,c2=-λ2+2b20βk2,c3=0,c4 is an arbitrary constant.

Hence, the Eq. (1) satisfies weak 1,2 condition and is a 2 order Briot-Bouquet differential equation. Obviously, the Eq. (1) satisfies the dominant condition. So, by Lemma 2, we know that all meromorphic solutions of the Eq. (1) belong to W. Now we will give the forms of all meromorphic solutions of the Eq. (1).

By (9), we infer that the indeterminant

Computer simulations for new solutions

In this section, we give some computer simulations to illustrate our main results. Here we take the new rational solutions w2r,2(z) and simply periodic solutions w2s,1(z) to further analyze their properties by the following Fig. 1, Fig. 2. Note that they have two distinct generation poles which are showed by these figures.

  • (1) Take k=1,z1=1,z0=0,β=112,λ=1 in w2r,2,

  • (2) Take z1=1,z0=0,β=-1,k=1,α=2,γ=3 in w2s,1,

Conclusions

Complex method is a very important tool in finding the traveling wave exact solutions of nonlinear evolution equations such as the Korteweg–de Vries (KdV) equation and the modified Benjamin–Bona–Mahony (mBBM) equation. In this paper, we employ the complex method to obtain all meromorphic exact solutions of complex Korteweg–de Vries (KdV) equation and the modified Benjamin–Bona–Mahony (mBBM) equation at first, then find out all traveling wave exact solutions of the Eqs. (KdV) and (mBBM). The

Acknowledgments

The second author would like to express his hearty thanks to Professors R. Conte, T. W. Ng, Fang Mingliang and Liao Liangwen for their helpful discussions and suggestions. This work is supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the first and second authors worked as visiting scholars. This work is also supported by the Visiting Academic Sponsor Project of Department of Mathematics and Statistics at Curtin University of Technology when

References (29)

Foundation item: Supported by the NSF of China (11271090), the NSF of Guangdong Province (S2012010010121) and the special fund of Guangdong Province and Chinese Ministry of Education on integration of production, education and research (2012B091100194).

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