Method for Solving the Sine-Gordon Equation

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

“…It is interesting to observe that the initial data (2) satisfies the advection equation u t + vu x = 0 with constant velocity v = µ/ε. In this sense, we may consider the initial data as being in uniform motion to the right with velocity v. Note, however, that if µ = 0, then for ε > 0 sufficiently small, the velocity v of the initial data exceeds the constraint |v| ≤ 1 imposed by the hyperbolic nature of the sine-Gordon equation (1). In this situation, one might expect the sine-Gordon equation to regularize the superluminal velocity of the initial data for t > 0 by some kind of catastrophic effect that destroys the profile of the initial data.…”

“…We consider the Cauchy problem in laboratory coordinates and we use the Riemann-Hilbert formulation of inverse scattering. For the sine-Gordon equation in characteristic coordinates, the inverse-scattering method was first given in [1] and [25]. The inverse-scattering method corresponding to the (noncharacteristic) Cauchy problem for the sine-Gordon equation in laboratory coordinates was worked out by Kaup [13].…”

“…In this paper, we consider the sine-Gordon equation (1) for all ε > 0 sufficiently small with the initial condition (2) sin f 2 := sech(x), cos f 2 := tanh(x), g := 2µ sech(x), x ∈ R where µ ∈ R is a parameter. We refer to the solution of the Cauchy problem as u(x, t; ε, µ).…”

“…The sine-Gordon equation (1) ε 2 u tt − ε 2 u xx + sin(u) = 0 describes a broad array of physical and mathematical phenomena. The partial differential equation (1) may be regarded as the continuum limit of a chain of pendula subject to an external (gravity) force and coupled to their nearest neighbors via Hooke's law.…”

Интегрируемые Системы И Квадраты Собственных Функций

“…Вскоре после этого Вадати [3] показал, что в ту же схему укладывается и модифицированное уравнение КдФ. Далее последовало решение уравнения синус-Гордон ОПР-методом, предложенное Абловицем, Каупом, Ньюэлом и Сегуром (АКНС) [4]. Вскоре после этого вышла классическая работа тех же авторов [5], в которой подчеркивалось, что ОПР похоже на нелинейное преобразование Фурье.…”