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The sine-Gordon equation in the semiclassical limit: by Robert J. Buckingham

By Robert J. Buckingham

The authors learn the Cauchy challenge for the sine-Gordon equation within the semiclassical restrict with pure-impulse preliminary information of enough energy to generate either high-frequency rotational movement close to the height of the impulse profile and likewise high-frequency librational movement within the tails. They express that for small instances self sustaining of the semiclassical scaling parameter, either kinds of movement are safely defined by means of specific formulae regarding elliptic services. those formulae show consistency with predictions of Whitham's formal modulation idea in either the hyperbolic (modulationally reliable) and elliptic (modulationally risky) circumstances

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Note, however, that the case when w is near P∞ must be considered separately; see [2] for details. 42 3. ELEMENTARY TRANSFORMATIONS OF J(w) With the notation of Y (w), T ∇ (w), and T Δ (w) established, we may write down the jump condition satisfied by M(w) across the various arcs of Σ in a simple form. 42) M+ (ξ) = σ2 M− (ξ)σ2 , ξ ∈ Σ>0 . 41) we used the fact that for ξ ∈ R+ , Q+ (ξ) + Q− (ξ) ≡ 0 and ΠN + (ξ)ΠN − (ξ) ≡ 1. Written this way, the jump relations display the key importance of the exponents 2iQ(ξ) + L(ξ) ± iθ0 (ξ), 2iQ(ξ) + ϕ∇ (ξ), and 2iQ(ξ)+ϕΔ (ξ).

ELEMENTARY TRANSFORMATIONS OF J(w) T ∇ (w)1/2 ≈ 1 and T Δ (w)1/2 ≈ 1 throughout the domain of definition of L∇ (w) and LΔ (w) respectively. 2. 51) Δ N+ (ξ)L∇ + (ξ) = N− (ξ)L− (ξ), ξ ∈ Σ∇Δ ∇ N+ (ξ)LΔ + (ξ) = N− (ξ)L− (ξ), ξ ∈ ΣΔ∇ . 52) Proof. Note that on Σ∇Δ and ΣΔ∇ we have Q+ (ξ) = Q− (ξ), L+ (ξ) = L− (ξ), g+ (ξ) = g− (ξ), Y+ (ξ) = Y− (ξ), and θ0+ (ξ) = θ0− (ξ). 33), a meaningful step since Σ∇Δ and ΣΔ∇ are disjoint from Z. 3. 53) ξ ∈ R+ . 57) AΔ (ξ) = O( N) and B Δ (ξ) = O λ2 N 2 N e−αλ/ N , λ = E− (ξ) > 0.

2 coincide. 31 32 3. ELEMENTARY TRANSFORMATIONS OF J(w) Δ = PN≺K : In this case we choose a < τ∞ < −1 and set Δ = PN≺K := PNK ∩ [a, τN ]. Thus Δ is localized near w = a. Δ = PNK : In this case we choose −1 < τ∞ < b and set Δ = PNK := PNK ∩ [τN , b]. Thus Δ is localized near w = b. ∇ = ∅: ∇= PN≺K : This is complementary to the case when Δ = ∅. In this case we choose a < τ∞ < −1 and set ∇ = PN≺K := PNK ∩ [a, τN ]. Thus ∇ is localized near w = a. ∇ = PNK : In this case we choose −1 < τ∞ < b and set ∇ = PNK := PNK ∩ [τN , b].

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