Abstract
In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.
Highlights
In this work, we study two equations of double dispersion in one and two dimensions
We look for low-order multipliers [19,22] as these provide us with physically interesting conservation laws
We study the (1+1)-dimensional and (2+1)-dimensional double dispersion equations, namely Equations (2) and (3)
Summary
The double dispersion (DD) equation arises in several physical applications. It is used in analyzing non-linear wave distribution in waveguide, interplay of waveguide and exterior medium, and, likelihood of energy interchange through lateral coverings of waveguide. Conservation laws have various utilizations in investigation of PDEs, for instance the determination of conserved quantities and the constants of motion. These can be applied to identify integrability and linearization and in verifying the correctness of numerical methods. The associated fourth-order non-linear ODEs for U are reduced to first-order variables separable equations by the application of conservation laws derived here for DD equations
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