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In mathematical physics, the **wave maps equation** is a geometric wave equation that solves

The **wave equation** is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

where is a connection.^{ [1] }^{ [2] }

It can be considered a natural extension of the wave equation for Riemannian manifolds.^{ [3] }

**Bessel functions**, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions *y*(*x*) of Bessel's differential equation

**Fractional calculus** is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D

The **Lotka–Volterra equations**, also known as the **predator–prey equations**, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

In the theory of partial differential equations, **elliptic operators** are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.

In mathematics, a **hyperbolic partial differential equation** of order *n* is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first *n* − 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is

In mathematics, **Fisher's equation** is the partial differential equation:

In physics, the **Fermi–Pasta–Ulam–Tsingou problem** or formerly the **Fermi–Pasta–Ulam problem** was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called **Fermi–Pasta–Ulam–Tsingou recurrence** – instead of ergodic behavior. One of the resolutions of the paradox includes the insight that many non-linear equations are exactly integrable. Another may be that ergodic behavior may depend on the initial energy of the system.

In mathematics, a **weak solution** to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.

The **Abel equation**, named after Niels Henrik Abel, is a type of functional equation which can be written in the form

The **Orr–Sommerfeld equation**, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are.

In mathematics, the **Dickson polynomials**, denoted *D _{n}*(

The **Ishimori equation (IE)** is a partial differential equation proposed by the Japanese mathematician Ishimori (1984). Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable Sattinger, Tracy & Venakides.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the **Hartree equations** for atoms, using the concept of *self-consistency* that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the **self-consistent field** method.

In physics, **fractional quantum mechanics** is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. This concept was discovered by Nick Laskin who coined the term *fractional quantum mechanics*.

The **fractional Schrödinger equation** is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term *fractional Schrödinger equation* was coined by Nick Laskin.

In physics, **Liouville field theory** is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.

In relativistic quantum mechanics and quantum field theory, the **Bargmann–Wigner equations** describe free particles of arbitrary spin *j*, an integer for bosons or half-integer for fermions. The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.

In mathematical physics, the **Whitham equation** is a non-local model for non-linear dispersive waves.

In the fields of dynamical systems and control theory, a **fractional-order system** is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Such systems are said to have *fractional dynamics*. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, power-law long-range dependence or fractal properties. Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in physics, electrochemistry, biology, viscoelasticity and chaotic systems.

**Monica Vișan** is a Romanian mathematician at the University of California, Los Angeles who has published highly cited work on the nonlinear Schrödinger equation.

- ↑ Tataru, Daniel (1 January 2005). "Rough solutions for the wave maps equation".
*American Journal of Mathematics*.**127**(2): 293–377. CiteSeerX 10.1.1.631.6746 . doi:10.1353/ajm.2005.0014. - ↑ Tataru, Daniel (2004). "The wave maps equation" (PDF).
*Bulletin of the American Mathematical Society (N.S.)*.**41**(2): 185–204. doi:10.1090/S0273-0979-04-01005-5. Zbl 1065.35199. - ↑ https://www.math.ucla.edu/~tao/preprints/wavemaps.pdf

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