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A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid. [12] At time t = 0 {\displaystyle t=0} the wall is made to move with constant speed U {\displaystyle U} in a fixed direction (for definiteness, say the x {\displaystyle x} direction and consider only the x − y {\displaystyle x-y} plane), one can ...
A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon equation [1] and the focusing nonlinear Schrödinger equation [2] are examples of one-dimensional partial differential equations that possess breather solutions. [3]
[12] An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259 [13] 20 examples of periodic solutions to the three-body problem. In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In ...
The most glaring example is the set of functions e 2πip · x ⁄ h. These are plane wave solutions of the Schrödinger equation for a free particle that are not normalizable, hence not in L 2. But they are nonetheless fundamental for the description. One can, using them, express functions that are normalizable using wave packets.
An instanton (or pseudoparticle [1] [2] [3]) is a notion appearing in theoretical and mathematical physics.An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory.
The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex ...
[1] [2] The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ε {\displaystyle \varepsilon } usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first ...
In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability ...