THE NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE FOKAS, ATHANASSIOS S.; HIMONAS, A. ALEXANDROU; MANTZAVINOS, DIONYSSIOS
Transactions of the American Mathematical Society,
01/2017, Volume:
369, Issue:
1
Journal Article
Peer reviewed
The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified ...transform method and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces on the half-line 0, ∞) for the spatial variable and on an interval 0, 𝑇, 0 < 𝑇 < ∞, for the temporal variable by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data (𝑢(𝑥, 0), 𝑢(0, 𝑡)) in
H
x
s
(
0
,
∞
)
×
H
t
(
2
s
+
1
)
/
4
(
0
,
T
)
, 𝑠 > 1/2, is established via a contraction mapping argument. In addition, this work places Fokas’ unified transform method for evolution equations into the broader Sobolev spaces framework.
The initial value problem for a system of modified Korteweg-deVries equations with data that are analytic on R and having uniform radius of analyticity r0 is studied. After proving an analytic ...version of known trilinear estimates in Sobolev spaces, local well-posedness is established and persistence of the radius of spatial analyticity is shown till some time T0. Then, for time t≥T0 it is proved that the radius of spatial analyticity is bounded from below by ct−(2+ε), for any ε>0.
It is shown that the initial value problem for the Fokas–Olver–Rosenau–Qiao equation (FORQ) is well-posed in Sobolev spaces Hs, s>5/2, in the sense of Hadamard. Furthermore, it is proved that the ...dependence on initial data is sharp, i.e. the data-to-solution map is continuous but not uniformly continuous. Also, peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in Hs for s<3/2.
Persistence of spatial analyticity is studied for periodic solutions of the dispersion-generalized KdV equation ut−|Dx|αux+uux=0 for α≥2. For a class of analytic initial data with a uniform radius of ...analyticity σ0>0, we obtain an asymptotic lower bound σ(t)≥ct−p on the uniform radius of analyticity σ(t) at time t, as t→∞, where p=max(1,4/α). The proof relies on bilinear estimates in Bourgain spaces and an approximate conservation law.
The
b
-Novikov equation is a one-parameter family of Camassa–Holm-type equations with cubic nonlinearities that possess multipeakon traveling wave solutions and for
b
=
3
gives the well known Novikov ...equation, which is integrable. Here, using appropriate two-peakon solutions, instability and nonuniqueness for the initial value problem of the
b
-Novikov equation is studied when the initial data belong in Sobolev spaces
H
s
,
s
<
3
/
2
, on both the line and the circle. The rectangular region of the
bs
-plane defined by
b
>
2
and
s
<
3
/
2
is divided into three subregions. The subregion that is below the line segment
s
=
2
-
b
4
,
2
<
b
<
4
, is characterized by the phenomenon of nonuniqueness. Then, to the right of this subregion the phenomenon of norm inflation occurs, which leads to instability and discontinuity of the solution map. However, on the segment
s
=
2
-
b
4
,
2
<
b
<
4
, either nonuniqueness or discontinuity may occur. All these are proved by constructing appropriate two-peakon solutions with arbitrary small initial size data that collide in arbitrary small time
T
. These solutions may become arbitrarily large near
T
. For
b
≤
2
, the two-peakon solutions do not work since there is no collision. Finally, it is well known that for
s
>
3
/
2
there is well-posedness no matter what is the value of
b
.
We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev ...exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index.
A higher dispersion KdV equation on the line Figueira, Renata; Himonas, A. Alexandrou; Yan, Fangchi
Nonlinear analysis,
October 2020, 2020-10-00, 20201001, Volume:
199
Journal Article
Peer reviewed
The Cauchy problem for a Korteweg–deVries equation with dispersion of order m=2j+1, where j is a positive integer, (KdVm), is studied with data in Sobolev and analytic spaces. First, optimal bilinear ...estimates in Bourgain spaces are proved and using them well-posedness in Sobolev spaces Hs, s>−j+14, is established. Then, well-posedness in analytic Gevrey spaces Gδ,s, δ>0, is proved by using an analytic version of the bilinear estimates. This implies that the uniform radius of analyticity persist for some time. For the later times a lower bound for the radius of spacial analyticity is derived, which is given by δ(t)≥ct−α, with α=43+ε, for any ε>0, when j=1, and α=1 when j≥2. Finally, it is shown that the regularity of the solution in the time variable is Gevrey of order m, and this is optimal.
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