Optimal Shape of an Underwater Moving Bottom Generating Surface Waves Ruled by a Forced Korteweg-de Vries Equation
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It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103–125, 1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg–de Vries equation to investigate the shape of the moving body capable of generating the highest first upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norms and compact supports. We establish analytically the existence of a maximizer saturating the norm constraint, derive the gradient of the functional, and then implement numerically an optimization algorithm yielding the desired optimal shape.
KeywordsShape optimization Existence theory Optimal control Surface wave generation Numerical simulation Forced Korteweg–de Vries equation Finite-difference methods
Mathematics Subject ClassificationPrimary 49K20 Secondary 35Q53 49M29 65M06 76B15 49J45 49J50
This work started in March 2010 under the original idea of E. Zuazua, who guided the author J. D. during an internship at the Basque Center for Applied Mathematics (Spain). J. D. gratefully acknowledges E. Zuazua for his support, dynamism, and guidance during this period, which now, looking back, feels like good old times. The work was then considerably improved while J. D. was finishing his Master degree at the Institut Elie Cartan de Lorraine (France), for which the author would like to acknowledge financial support and thank A. Henrot for the encouragement to pursue this study. R. B. acknowledges the support of Science Foundation Ireland under grant 12/IA/1683. Finally, we sincerely thank the Editors and anonymous Referees for their valuable comments, which helped us improve the quality of this manuscript.
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