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Benjamin–Ono Equation
In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by and . The Benjamin–Ono equation is :u_t+uu_x+Hu_=0 where ''H'' is the Hilbert transform. See also * Bretherton equation In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: :u_+u_+u_+u = u^p, with p integer and p \ge 2. While u_t, u_x and u_ denote partial derivatives of the scalar fiel ... References * * External linksBenjamin-Ono equations: Solitons and Shock Waves Nonlinear partial differential equations Integrable systems {{theoretical-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integro-differential Equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form : \fracu(x) + \int_^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0. As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation. Example Consider the following second-order problem, : u'(x) + 2u(x) + 5\int_^u(t)\,dt = \theta(x) \qquad \text \qquad u(0)=0, where : \theta(x) = \left\{ \begin{array}{ll} 1, \qq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Wave
Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance (as in the case of the thermocline in lakes and oceans or an atmospheric inversion), the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid. Internal waves, also called internal gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude, and influence of external forces. If propagating horizontally along an interface where the density rapidly decreases with height, they are specifically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the function 1/(\pi t) (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal . The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions. Definition The Hilbert transform of can be thought of as the convolution of with the function , known as the Cauchy kernel. Because is not integrable across , the integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bretherton Equation
In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: :u_+u_+u_+u = u^p, with p integer and p \ge 2. While u_t, u_x and u_ denote partial derivatives of the scalar field u(x,t). The original equation studied by Bretherton has quadratic nonlinearity, p=2. Nayfeh treats the case p=3 with two different methods: Whitham's averaged Lagrangian method and the method of multiple scales. The Bretherton equation is a model equation for studying weakly-nonlinear wave dispersion. It has been used to study the interaction of harmonics by nonlinear resonance. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions. Variational formulations The Bretherton equation derives from the Lagrangian density: : \mathcal = \tfrac12 \left( u_t \right)^2 + \tfrac12 \left( u_x \right)^2 -\tfrac12 \left( u_ \right)^2 - \tfrac12 u^2 + \tfrac u^ through the Euler–Lagrange equation: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equations
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
