Creation of a new model for water waves and its mathematical analysis

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K18742
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Keio University
• Principal Investigator: IGUCHI Tatsuo 慶應義塾大学, 理工学部(矢上), 教授 (20294879)
• Project Period (FY): 2017-06-30 – 2022-03-31
• Keywords: 水の波 / 変分構造 / 適切性 / 浅水波近似 / 磯部‐柿沼モデル / Hamilton構造 / 孤立波 / 極限波

Outline of Final Research Achievements

Starting from the basic equations for water waves, we derived a model, called the Isobe-Kakinuma model, for water waves by using a variational structure of the basic equations, and analyzed structures of the model from mathematical point of view. More precisely, we showed that the initial value problem of the model is well-posed locally in time, that the model is a higher order shallow water approximation, that the model possesses a Hamiltonian structure, and so on. Moreover, by numerical analysis, we investigate the global structure of solitary wave solutions of the model and observed that the existence of a solitary wave of extreme form with a maximal height and a sharp crest.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

水の波の基礎方程式系は非常に複雑な偏微分方程式系であるため，その解の構造の理解や数値シミュレーションを目的として非常に多くの近似モデルが提唱されている．本研究により，磯部‐柿沼モデルはこれまでに提唱されているどの近似モデルよりも，ある意味で優位なモデルであることを数学的に保証することができた．特に，孤立波解が限界波高をもつ近似モデルは，トイモデルを除いて磯部‐柿沼モデル以外に見い出されていないことは特筆すべきである．