Stability of nonlinear waves in integrable Hamiltonian PDEs Dmitry Pelinovsky Department of Mathematics, McMaster University, Ontario, Canada http://dmpeli.math.mcmaster.ca Workshop "Linear and Nonlinear Dirac Equations" Como, Italy, February 8-10, 2017 Dmitry Pelinovsky (McMaster University) Stability of nonlinear waves 1 / 32
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Stability of nonlinear waves in integrable Hamiltonian PDEs · stability of solitary waves can be obtained for the NLS equation. Mizumachi–P. (2012); Cuccagna–P. (2014); Contreras–P
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Stability of nonlinear waves
in integrable Hamiltonian PDEs
Dmitry PelinovskyDepartment of Mathematics, McMaster University, Ontario, Canada
http://dmpeli.math.mcmaster.ca
Workshop "Linear and Nonlinear Dirac Equations"Como, Italy, February 8-10, 2017
Many classical PDE problems, which were opened in the functional-analyticframework, have been recently solved for the integrable nonlinear PDEs.
Example 1 : Global existence for the derivative NLS equation
iut + uxx + i(|u|2u)x = 0, t > 0,u|t=0 = u0 ∈ X ,
where X is some Banach space.
Definition
The Cauchy problem is locally well-posed in X if there exists an uniquesolution u(t, ·) ∈ X for t ∈ (−T ,T ) with finite T > 0 and the solutionmap u0 7→ u(t, ·) is continuous. It is globally well-posed if T can bearbitrarily large.
Tsutsumi & Fukuda (1980) established local well-posedness in Hs(R)with s > 3
2and extended solutions globally in H2(R) for small data in
H1(R)
Hayashi (1993) used gauge transformation of DNLS to a system ofsemi-linear NLS and established local and global well-posedness inH1(R) under the constraint ‖u0‖L2 <
√2π.
Global existence was proved in Hs(R) for s > 32
33(Takaoka, 2001),
s > 1
2(Colliander et al, 2002), and s = 1
2(Mio-Wu-Xu, 2011) under
the same constraint ‖u0‖L2 <√
2π.
Recent development:global existence without restriction on the L2(R) norm.Liu–Perry–Sulem (2016); P–Shimabukuro (2017).
Example 2 : Orbital stability in spaces of low regularity
Orbital stability in H1(R) is proved with the energy method(Lyapunov functions and constrained minimization)Weinstein (1985), Shatah–Strauss (1985), Grillakis et al. (1987).
Energy methods do not work in L2(R) due to lack of control.
With the Bäcklund–Darboux transformation, orbital and asymptoticstability of solitary waves can be obtained for the NLS equation.Mizumachi–P. (2012); Cuccagna–P. (2014); Contreras–P (2014).
N-soliton solutions are orbitally stable in HN(R) KdV [Sachs - Maddocks (1993)] NLS [Kapitula (2006)] Derivative NLS [Le Coz–Wu (2016)]
Breathers are orbitally stable in H2(R) modified KdV [Alejo–Munoz (2013)] sine-Gordon [Alejo–Munoz (2016)]
In the rest of my talk, I will restrict attention to stability of relativeequilibria in Hamiltonian systems (solitary waves, periodic waves) by usingenergy methods.
N-soliton solutions are orbitally stable in HN(R) KdV [Sachs - Maddocks (1993)] NLS [Kapitula (2006)] Derivative NLS [Le Coz–Wu (2016)]
Breathers are orbitally stable in H2(R) modified KdV [Alejo–Munoz (2013)] sine-Gordon [Alejo–Munoz (2016)]
In the rest of my talk, I will restrict attention to stability of relativeequilibria in Hamiltonian systems (solitary waves, periodic waves) by usingenergy methods.
Orbital Stability Theorem [Grillakis–Shatah–Strauss (1990)]
Assume no symmetries/zero eigenvalues of H ′′(u0). If H ′′(u0) has nonegative eigenvalues, then JH ′′(u0) has no unstable eigenvalues and u0
is linearly and nonlinearly stable.
Assume zero eigenvalue of H ′′(u0) of multiplicity N and related N
symmetries/conserved quantities. If H ′′(u0) has no negativeeigenvalues under N constraints, then JH ′′(u0) has no unstableeigenvalues and u0 is orbitally stable.
Orbital Stability Theorem [Grillakis–Shatah–Strauss (1990)]
Assume no symmetries/zero eigenvalues of H ′′(u0). If H ′′(u0) has nonegative eigenvalues, then JH ′′(u0) has no unstable eigenvalues and u0
is linearly and nonlinearly stable.
Assume zero eigenvalue of H ′′(u0) of multiplicity N and related N
symmetries/conserved quantities. If H ′′(u0) has no negativeeigenvalues under N constraints, then JH ′′(u0) has no unstableeigenvalues and u0 is orbitally stable.
Critical points of H + ωQ for a fixed ω ∈ (−1, 1) satisfy the stationaryMTM equations. After the reduction (u, v) = (U,U), we obtain thefirst-order equation
idU
dx− ωU + U = 2|U|2U.
The MTM soliton U = Uω satisfies the first-order equation.
Critical points of R +ΩQ for some fixed Ω ∈ R satisfy another systemof equations. After the reduction (u, v) = (U,U), we obtain thesecond-order equation
d2U
dx2+ 6i |U|2 dU
dx− 6|U|4U + 3|U|2U + U3 = ΩU.
U = Uω also satisfies the second-order equation if Ω = 1 − ω2.
Critical points of H + ωQ for a fixed ω ∈ (−1, 1) satisfy the stationaryMTM equations. After the reduction (u, v) = (U,U), we obtain thefirst-order equation
idU
dx− ωU + U = 2|U|2U.
The MTM soliton U = Uω satisfies the first-order equation.
Critical points of R +ΩQ for some fixed Ω ∈ R satisfy another systemof equations. After the reduction (u, v) = (U,U), we obtain thesecond-order equation
d2U
dx2+ 6i |U|2 dU
dx− 6|U|4U + 3|U|2U + U3 = ΩU.
U = Uω also satisfies the second-order equation if Ω = 1 − ω2.
Periodic waves are not constrained minimizers of neither E nor R
if the period of perturbations is multiple to the period of waves.
Nevertheless, there exists a range of values for parameter c such that theenergy functional Λc := R − cE is positively definite at u0.[Bottman–Deconinck–Nivala (2011)]
Theorem (Gallay–P (2015))
For all E ∈ (0, 1), the second variation of Λc at the periodic wave u0 is
nonnegative for perturbations in H2
Nper only if c ∈ [c−, c+] with
c± := 2 ± 2κ
1 + κ2, κ =
√
1 − E1 + E .
Moreover, it is strictly positive up to symmetries in (c−, c+) if E is small.
Algorithmic search of the commuting operatorWe are looking for an operator Mc,p to satisfy the commutability relation
Lc,p∂xMc,p = Mc,p∂xLc,p.
Since 1D periodic waves u = v(x + ct) are also critical points of R(u), theHessian operator Mc,p related to R(u) satisfies this commutability relation.The operator Mc,p is given by
Mc,p = ∂4
x + 10∂xv(x)∂x − 10cv(x)− c2
−10
3p2
(
1 + v(x)∂−2
x + ∂−1
x v(x)∂−1
x + ∂−2
x v(x))
+5
9p4∂−4
x .
Lemma
For every p 6= 0, no value of b ∈ R exists such that Mc,p − bLc,p is
positive. Moreover, the number of negative eigenvalues quickly grows in
Spectral stability theory is well-developed for relative equilibria inHamiltonian systems, when the Hessian operators have finitely manynegative eigenvalues.
Orbital stability holds in Hamiltonian systems if the relativeequilibrium is a non-degenerate minimum of energy under constraintsof fixed mass and momentum.
For many integrable PDEs (MTM, NLS, KdV), one can usehigher-order Hamiltonians to conclude on orbital stability of nonlinearwaves.
For the KP-II equation (in 2D), one can find positive-definite operatorunrelated to conserved quantities in order to conclude on spectralstability of nonlinear waves.