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On similarity in the evolution of semilinear wave and Klein-Gordon equations: Numerical surveys
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10.1063/1.3684983
/content/aip/journal/jmp/53/2/10.1063/1.3684983
http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/2/10.1063/1.3684983

Figures

Image of FIG. 1.
FIG. 1.

Blowup out of the origin for NLW n = 3, p = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by U 0 profile. From fit U(t, r max )−3a(Tt) we get the blowup time T = 0.0404829 and a = 1.49855, which approximately equals .

Image of FIG. 2.
FIG. 2.

Blowup out of the origin for NLKG n = 3, p = 7. Upper panel: Normalized spatial profiles . Bottom panel: Near-blowup asymptotics is approximated by U 0 profile of NLW providing that Conjecture 1 is true. From fit U(t, r max )−3a(Tt) we get the blowup time T = 0.0405039 and a = 1.49994 which approximately equals .

Image of FIG. 3.
FIG. 3.

Central blowup for NLW n = 3, p = 7. Upper panel: Near-blowup asymptotics is approximated by . From the fit c 1 = 1.20094, T = 3.18068. Bottom panel: Spatial profile near the blowup with the same T, c 1 and δ = 0.001154.

Image of FIG. 4.
FIG. 4.

Central blowup for NLKG n = 3, p = 7. Upper panel: Near-blowup asymptotics, as Conjecture 1 suggest, is approximated by . From the fit c 1 = 1.51724, T = 3.15336. Bottom panel: Spatial profile near the blowup with the same T, c 1 and δ = 0.00337.

Image of FIG. 5.
FIG. 5.

Upper panel: Sketch of initial parameters plane for critical NLW n = 3, p = 5; s = 2. Edge 2 corresponds to the type II blowup and edge 4 to the static solution. Area 3 presents dispersion and areas 1 and 5 generic blowup. Near edge 4 there is a dispersion or generic/ODE blowup after a departure from the static intermediate attractor and near edge 2 there is a dispersion or generic/ODE blowup after departure from the type II blowup intermediate attractor. Scanning with 10−6 resolution does not reveal any exotic structures of the edges, e.g., fractals, etc. The cusp of 4 is located around the point (x 0; A) ≈ (0.5634; 0.2025). Bottom panel: The left figure presents U(t, r = 0) near edge 4. There is plateau that corresponds to the vicinity of the static solution. The right figure presents the solutions at r = 0 near edge 2 of the type II blowup. There is no static episode in the evolution, and instead, both solutions before the departure approach the type II blowup solution.

Image of FIG. 6.
FIG. 6.

Upper panel: Convergence of the solution of critical NLW to a normalized static one for the type II blowup for n = 3, p = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(t) = 1/U(t, r = 0)2. Bottom panel: The departure from the type II intermediate attractor. One can note that there exist two times of a blowup. The type II intermediate attractor has the blowup time T 1, but before that time the unstable mode takes over, and the solution blows up in a generic/ODE way with the blowup time T 2 < T 1.

Image of FIG. 7.
FIG. 7.

Upper panel: Convergence of the solution of critical NLKG to normalized wave static one for blowup for n = 3, p = 5 in the intermediate region. Snapshots of the profiles for a given time are scaled , where λ(t) = 1/U(t, r = 0)2. Bottom panel: Corresponding values of U(t, r = 0)−2.

Image of FIG. 8.
FIG. 8.

Upper panel: Convergence of the solution of supercritical NLW to u 1 profile in the transient phase for n = 3, p = 7. From theory it is known that . Bottom panel: The departure from the intermediate attractor u 1. Near intermediate attractor .

Image of FIG. 9.
FIG. 9.

Upper panel: Convergence of the NLKG solution to the NLW u 1 profile in the transient phase for n = 3, p = 7. From Conjecture 1, it is supposed that as for NLW. Bottom panel: The departure from the NLW intermediate attractor u 1. Near intermediate attractor .

Image of FIG. 10.
FIG. 10.

Upper panel: Convergence of the solution of supercritical NLW to u 1 profile in the transient phase for n = 4, p = 5. From theory it is known, that . Bottom panel: In the intermediate region (Tt)1/2 U(t, r/(Tt)) ≈ u 1(ρ). One can see that the intermediate attractor of NLW matches u 1 also at ρ = 1.

Image of FIG. 11.
FIG. 11.

Upper panel: Convergence of the NLKG solution to the NLW u 1 profile in the transient phase for n = 4, p = 5. From Conjecture 1, it is supposed that as for NLW. Bottom panel: As for NLW in the intermediate region (Tt)1/2 U(t, r/(Tt)) ≈ u 1(ρ). As previously the intermediate attractor of NLKG matches u 1 also at ρ = 1.

Tables

Generic image for table
Table I.

Eigenvalues for a few unstable modes of u 1 for supercritical NLW. The values of u 1(ρ = 0) are also presented. They were obtained using the numerical methods described in Ref. 18. One can observe that the eigenvalues decrease when n and p increase. Therefore, for higher n and p the unstable solutions “lives” longer, because the unstable mode grows more slowly. In this paper, we restricted ourselves to odd p. But we think it is appropriate to indicate the same structure for supercritical even p, as well as for critical even p in Table II.

Generic image for table
Table II.

Eigenvalues for unstable modes v 1 for critical NLW. Here . Similarly as for subcritical cases, the eigenvalues are smaller for larger n values.

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/content/aip/journal/jmp/53/2/10.1063/1.3684983
2012-02-22
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: On similarity in the evolution of semilinear wave and Klein-Gordon equations: Numerical surveys
http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/2/10.1063/1.3684983
10.1063/1.3684983
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