The sine-Gordon equation in the semiclassical limit: Dynamics of fluxon condensates

Abstract

We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases. Contents 1 arXiv:1103.0061v1 [math-ph] 1 Mar 2011Using ω = v p k together with the nonlinear dispersion relation in the form (1.22) shows that k and ω may be eliminated in favor of v p and E:where σ = ±1 is an arbitrary sign whose role is to select different branches of the dispersion relation. Therefore, the variational modulation equation (1.23) becomesand the conservation of waves equation (1.11) becomesThe proof of this proposition is a rather straightforward application of Fubini's Theorem and is given in Appendix A.We will require that the WKB phase integral have certain analyticity properties to be outlined in Proposition 1.2 below. We now make an assumption on G that will be sufficient to establish Proposition 1.2 and that can easily be checked for a given G:Assumption 1.4. The function G is strictly increasing and real-analytic at each x > 0, and the positive and real-analytic functioncan be analytically continued to neighborhoods of m = 0 and m = G(0) 2 , with G (0) > 0 and G (G(0) 2 ) > 0.We point out that the class of functions G (m) satisfying Assumption 1.4 obviously parametrizes a corresponding class of admissible functions G(x) by simply viewing (1.48) as an equation to be solved for x = G −1

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

“…This cubic polynomial satisfies the following: The lemma is stated in theorem 1 in [11] (with different notation: here we only note that the right-hand side of (6.3) is an analytic function in D c because .´/ has a jump discontinuity across B whereby C D ). We also refer to section 4.3 in [10] for another presentation.…”

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

“…As Corollary 2 is formulated, the points t = t a and t = t b appear to present an obstruction to continuation of the plane-wave approximation to positive x from the boundary. While the details are not easy to explain, the fact is that the complex phase function g can indeed be continued away from the boundary near such points, which become curves t = t a (x) and t = t b (x) for x > 0 along which one of the band endpoints is fixed to an endpoint of the interval [k a , k b ] (see, for example, Section 4.3.2 of [20]). There still remains, however, a technical issue in that it appears that a new type of inner parametrix is required near the corresponding point in the k-plane, and to our knowledge this has not been worked out.…”

“…Consider the second equation of the system (20) at x = 0 along with (22) in the formal semiclassical limit → 0, assuming that η(0, t) = 0. This means that we simply neglect the terms explicitly proportional to or 2 in each case, yielding the formal approximations: ∂σ ∂t (0, t) + u(0, t) 2 + 2η(0, t) 2 ≈ 0 and − i Q N (t)…”

“…The key point of our approach is that by neglecting the formally small dispersive terms in (20) we obtain a system that is first-order in x and hence allows the unknown Neumann data to be explicitly eliminated in favor of t-derivatives that may be computed along the boundary from the given Dirichlet data. This approximation is a purely local relation between the two boundary values, and in particular the approximate Dirichlet-to-Neumann map is independent of initial data q 0 .…”

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

“…The matrix Baker-Akhiezer function will be useful for applying to Cauchy problems with periodic (quasi-periodic) finite-gap initial data as well as for the initial-boundary value problems with such type of initial and boundary functions. The suggested RH problem will be also useful for studying the long time/large space asymptotic behavior of solutions of different initialboundary value problems to the MB equations by the way as, for example, in [6,7,8,9,10,11,12,13,14,15,16,17,18,21,22,26,36,37,38,43,44,56,57,59,60]. The focusing nonlinear Schrödinger equation and its finite-gap solutions are widely used for modeling of the so-called rogue waves.…”