코테베그-드 브리스 방정식(KdV equation)

개요

any localized nonlinear wave which interacts with another (arbitrary) local disturbance and always regains asymptotically its exact initial shape and velocity (allowing for a possible phase shift)

Using a wave tank, he demonstrated four facts
- First, solitary waves have a hyperbolic secant shape.
- Second, a sufficiently large initial mass of water produces two or more independent solitary waves.
- Third, solitary waves cross each other “without change of any kind.”
- Finally, a wave of height h and traveling in a channel of depth d has a velocity given by the expression (where g is the acceleration of gravity), implying that a large amplitude solitary wave travels faster than one of low amplitude.

\(u(x,t)=f(x-ct)\)로 두자.

\(f'''= 6ff'-cf'\)

\(f''=3f^2-cf+b\)

\(f''f'=(3f^2-cf+b)f'\)

\(\frac{1}{2}(f')^2=f^3-\frac{c}{2}f^2+bf+a\)

1844 러셀이 관찰과 실험을 통해 솔리톤을 발견
1895 코테베그와 드 브리스가 1-솔리톤의 해석적 해를 구함
- 러셀의 발견을 모형화하고 미분방정식을 도입
1965 자부스키와 크루스칼의 수치해석적 연구
- 두 솔리톤(1-soliton)의 상호작용
- 크기가 다른 두 솔리톤이 깔끔하게 상호작용한다는 사실을 발견
John Scott Russell and the solitary wave
수학사 연표