Download e-book for kindle: Small divisor problem in the theory of three-dimensional by Gerard Iooss, Pavel I. Plotnikov

By Gerard Iooss, Pavel I. Plotnikov

ISBN-10: 0821843826

ISBN-13: 9780821843826

The authors think of doubly-periodic vacationing waves on the floor of an infinitely deep excellent fluid, in basic terms subjected to gravity g and caused by the nonlinear interplay of 2 easily periodic touring waves making an attitude 2[theta] among them. Denoting via [mu] = gL/c2 the dimensionless bifurcation parameter (L is the wave size alongside the path of the vacationing wave and c is the rate of the wave), bifurcation happens for [mu] = cos[theta]. For non-resonant situations, we first supply a wide relations of formal three-d gravity traveling waves, within the kind of a variety in powers of the amplitudes of 2 easy vacationing waves. 'Diamond waves' are a specific case of such waves, after they are symmetric with recognize to the course of propagation. the most item of the paper is the facts of life of such symmetric waves having the above pointed out asymptotic growth. as a result occurence of small divisors, the most hassle is the inversion of the linearized operator at a non trivial aspect, for employing the Nash Moser theorem. This operator is the sum of a moment order differentiation alongside a definite path, and an integro-differential operator of first order, either based periodically of coordinates. it really is proven that for the majority angles [theta], the three-d vacationing waves bifurcate for a collection of 'good' values of the bifurcation parameter having asymptotically an entire degree close to the bifurcation curve within the parameter airplane ([theta], [mu])

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New PDF release: Small divisor problem in the theory of three-dimensional

The authors give some thought to doubly-periodic traveling waves on the floor of an infinitely deep excellent fluid, simply subjected to gravity g and because of the nonlinear interplay of 2 easily periodic touring waves making an attitude 2[theta] among them. Denoting by way of [mu] = gL/c2 the dimensionless bifurcation parameter (L is the wave size alongside the path of the traveling wave and c is the speed of the wave), bifurcation happens for [mu] = cos[theta].

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32) V1 (X(Z))−3 |V (X(Z))|2 + V (X(Z))·∇X η(X(Z)) 2 1/2 a(X(Z))∂z2 Z(Z) −1 , has symmetry (e, e), and is close to µ. 6. This equation can be solved as follows. We first note that it gives a unique d2 with symmetry (o, e) provided that π 1 q1/2 (Z)dz1 . 34) π/τ π −π/τ −π ν = 2πτ q1/2 (Z)dz1 −2 dz2 , which is just 1/µ when (ψ, η) = 0. 33) by the change z2 → z2 + π/τ. This means that e2 is indeed π/τ − periodic. 6 that ||q − µ||C m−4 ≤ cm (||U ||4 )||U ||m, |ν − 1/µ| ≤ c||U ||4 , ||d2 ||C m−4 + ||e2 ||C m−4 ≤ cm (||U ||4 )||U ||m.

23) ||G−1 u||r ≤ cε||u||r−1, for 1 ≤ r ≤ ρ − 9, ||G−1 u||s ≤ c(ε||u||s−1 + El ||u||0 ), for s ≤ l − 9. Moreover, operators G1 , G0 and G−1 satisfy the symmetry properties Gj u(±Y ∗ ) = Gj u∗ (±Y ), j = 1, 0, −1, u∗ (Y ) = u(Y ∗ ). Proof. The proof is given in Appendix G. The formula for the principal term G1 is a classic result of the theory of pseudodifferential operators [Ho]. The expression for the second term in local Riemann coordinates was given in [ABB]. 20) are new. Note that the ratio det G/J 2 = 1+|∇X η(X)|2 is a scalar invariant and the real part of G0 can be rewritten in the invariant form 1 LG − 2M F + N F 1 Lξ12 + 2M ξ1 ξ2 + N ξ22 J − .

It is easy to see that ω1 2−j−4 < 4q/(3ω1 n4 ) for all j > ς(n). 39) that for all such j, the intervals Ij (m, n) have nonempty intersections with I∞ (m, n) and ∪j>ς(n) Ij (m, n) ⊂ I˜∞ (m, n), which yields meas j>ς(n) 20q 1 Ij (m, n) ≤ meas I˜∞ (m, n) ≤ , 3ω1 n4 where I∞ (m, n) is the 4q/(3ω1 n4 )- neighborhood of I∞ (m, n). 11, the intersection Ij (m, n) ∩ [0, r] is empty for n < c3 r−1/(3+α) , we have ℘(r) ≤ n≥c3 r −1/(3+α) c n4 n≥c3 r −1/(3+α) ς(n) card {m : Ij (m, n) ∩ [0, r] = ∅}+ j=1 c card {m : I∞ ∩ [0, r] = ∅}, n4 4.

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Small divisor problem in the theory of three-dimensional water gravity waves by Gerard Iooss, Pavel I. Plotnikov


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