weiguo rui, yao long and bin he- periodic wave solutions and solitary cusp wave solutions for a...

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Rostock. Math. Kolloq. 61, 57–71 (2006) Subject Classiﬁcation (AMS) 37K40, 37K50 Weiguo Rui 1 , Yao Long and Bin He Peri odic wave sol uti ons and sol itary cus p wave sol u- tions for a higher order wave equation of KdV type 2 ABSTRACT. This paper is the continuation of Ref. [1]. Both the bifurcatio n theory of planar dynamical system and elliptic function integral method are applied to study a higher order wave equation of KdV type. And the parametric space is redivided when the integral constant g = 0. Man y explicit and implicit solution s of periodic wav e and solita ry cusp wave are obtained. KEY WORDS. higher order wave equation of KdV type, solitary cusp wave solution, periodic wave solution, elliptic function integral method. 1 In tr oduction In this paper, we will seek periodic wave and solitary cusp wave solutions for the following higher order wave equation of KdV type (see [ 1, 2]): u t + u x + αuu x + βu xxx + α 2 ρ 1 u 2 u x + αβ (ρ 2 uu xxx + ρ 3 u x u xx ) = 0 , (1.1) where ρ i (i = 1, 2, 3) are free parameters and α, β are positive real constants which character- ize, respectivel y , the long wa vel ength and short amplitude of the wave s. Just as Tzirtzilakis, E. [2] said, the equation (1.1) is a water wave equation of KdV type which is more physically and practically meaningful. By the local coordinate transformation u = v − αρ 1 v 2 − β 3ρ 1 + 7 4 ρ 2 − 1 2 ρ 3 v xx , (1.2) Eq. (1.1) can be transformed into the following simple equation, see [ 3, 4]: v t − 3 2 βρ 2 v xxt + β (1 − 3 2 ρ 2 )v xxx + αvv x − 1 2 αβ ρ 2 (vv xxx + 2v x v xx ) = 0 , (1.3) 1 Corresponding author: E-mail address: [email protected] 2 This research was supported by Natural Science Foundation of Honghe University (XJZ10401) and the National Natural Science Foundation of China (10261008).

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Page 1: Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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Page 3: Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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8/3/2019 Weiguo Rui, Yao Long and Bin He- Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type

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