Abstract
A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.
Highlights
IntroductionThe Whitham-Broer-Kaup model equations (WBK) [1]-[5] in the shallow water small-amplitude regime are that ut + uux + vx + β uxx =0,
The Whitham-Broer-Kaup model equations (WBK) [1]-[5] in the shallow water small-amplitude regime are that ut + uux + vx + β uxx =0, (1) vt + x − β vxx + α uxxx = 0, (2)where u = u ( x,t ) represents the horizontal velocity, and v = v ( x,t ) the height deviated from the equilibrium position of the liquid, α and β are constants
The original homogeneous balance method (HB) is simplified by using a logarithmic function instead of the undetermined function appearing in the original HB
Summary
The Whitham-Broer-Kaup model equations (WBK) [1]-[5] in the shallow water small-amplitude regime are that ut + uux + vx + β uxx =0,. In the latest paper [6], the multiple soliton solutions of Equations (1) and (2) have been obtained by using the simplified form of Hirota’s direct method. (2014) Simplified Homogeneous Balance Method and Its Applications to the Whitham-Broer-Kaup Model Equations. We will apply a simplified homogeneous balance method to investigate the WBK model Equations (1) and (2), by this method a nonlinear transformation that from the solution for a linear equation to the solution for the WBK model equations is derived, and more type of solutions than those given in [6] are obtained via the nonlinear transformation successfully
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