Kortewegde

Kortewegde is a term sometimes used to refer to the Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation that arises in the study of long waves in shallow water and other dispersive media. The name honors Dutch physicists Diederik Johannes Korteweg and Gustav de Vries, who derived the equation in 1895 to model the balance between nonlinear steepening and dispersive spreading of waves.

Form and key features: In its common dimensionless form, the KdV equation is u_t + 6 u u_x + u_{xxx} = 0, where u(x,t) represents wave amplitude, and subscripts denote partial derivatives. The equation captures a nonlinear advection term, a dispersive term, and their interaction. It is integrable, admitting exact solutions and an infinite set of conserved quantities. A notable property is the existence of solitons—robust, localized waves that maintain shape and speed and can interact with other solitons and emerge unchanged apart from phase shifts.

Origins and impact: The KdV equation was introduced to describe shallow water waves but has since become a central model in nonlinear wave theory. Its study contributed to the development of soliton theory and the inverse scattering transform, with foundational demonstrations of soliton behavior occurring in numerical and analytical work during the mid-20th century.

Applications and relevance: Beyond fluid dynamics, the KdV equation appears in contexts such as plasma physics, lattice dynamics, and nonlinear optics, particularly under conditions that approximate weak nonlinearity and dispersion. It remains a canonical example of an integrable nonlinear system and a benchmark for techniques in mathematical physics.

Terminology: While "Kortewegde" is used in some texts as shorthand, the standard reference remains the Korteweg–de Vries (KdV) equation.