Complete classification for the (1+1)D non-linear boussinesq equations

Abstract

This thesis contributes to the knownanalytical solutions of the (1+1)D non-linearBoussinesq (NB) equations, which effectively describe solitary wave motion in shallow water (typically used to describe tsunami wave motion) using a systematic symmetry approach. These results can be used to enrich the analytical validation ofnumerical models for tsunami detection systems. Tsunami detection systems based on numerical models provide real-time forecasting and inundation mapping, playing a critical role in evacuation plans. These models have become more sophisticated over the past three decades through precise and rigorous analytical and experimental validation. Although analytical and experimental benchmarking cannot ensure that numerical models that pass benchmark tests will always produce realistic inundation predictions, validation largely reduces the uncertainty in the results to initial geophysical conditions [1]. The true benefit of analytical solutions in tsunami model predictions is the ability to identify the dependence of the desired results (such as run-up height, wave height and wave speed) on the independent variables of the model (such as water depth and beach slope). Problem scaling is also made simpler through analytical calculations, as opposed to numerical techniques, which usually require repeated numerical computations. Lastly, by comparing the numerical solutions to exact analytical solutions, systematic errors can be identified, making analytical solutions useful in validating complex numerical models with realistic applications [1]. The use of conserved quantities as an analytical validation mechanism is a good example of how analytical solutions can assist with numerical tsunami model validation. Upon substitution of these quantities into the numerical models, these conserved quantities should remain constant throughout the calculation. Although analytical solutions are powerful for the reasons previously stated, these solutions are complex to obtain, and therefore, not as widely used in comparison to experimental validation for tsunami detection system development. The (1+1)DBoussinesq equations investigated in this work are considered for the equa-

tions’ ability to model tsunamis via solitary wave motion. Three models are investigated, namely the (1+1)DBoussinesq equations for amoving bathymetry, the (1+1)D linearBoussinesq (LB) equations at a constant water depth and the NB equations at a constant water depth. We obtain three cases of solutions for the NB equations at a constant water depth dependent on the wave speed c 2, namely (i) 0 < c 2 < 1, c 2 6= 1 15 and c 2 6= 1 3 ; (ii) c 2 = 1 15 and (iii) c 2 = 1 3 . To our knowledge, the invariant exact travelling wave solutions for the NB equations for cases (i) and (iii) are new and can be used in analytical validation of numerical models for tsunami detection systems, whilst the remaining case, case (ii), is identical to that displayed in the work of Dutykh and Dias [2]. We provide physically realistic explanations of the results relating to the horizontal component of the fluid velocity and surface elevation of the solitary wave for the invariant exact solutions. This is done by considering seven cases dependent on the wave speed c 2, namely 0 < c 2 < 1 15 , c 2 = 1 15 , 1 15 < c 2 < 1 3 , c 2 = 1 3 , 1 3 < c 2 < 1, c 2 = 1 and 1 < c 2 <1. To our knowledge, these are newly discovered solutions and results.