Nonlinear Dyn
https://doi.org/10.1007/s11071-018-4373-0
ORIGINAL PAPER
Dynamics of high-order solitons in the nonlocal nonlinear
Schrödinger equations
Bo Yang · Yong Chen
Received: 15 February 2018 / Accepted: 14 May 2018
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract A study of high-order solitons in three non-
local nonlinear Schrödinger equations is presented.
These include the PT -symmetric, reverse-time, and
reverse-space-time nonlocal nonlinear Schrödinger
equations. General high-order solitons in three different
equations are derived from the same Riemann–Hilbert
solutions of the AKNS hierarchy, except for the differ-
ence in the corresponding symmetry relations on the
“perturbed” scattering data. Dynamics of general high-
order solitons in these equations is further analyzed.
It is shown that the high-order fundamental-soliton is
moving on several different trajectories in nearly equal
velocities, and they can be nonsingular or repeatedly
collapsing, depending on the choices of the parame-
ters. It is also shown that the high-order multi-solitons
could have more complicated wave structures and
behave very differently from high-order fundamental-
solitons. More interestingly, via the combinations of
different size of block matrix in the Riemann–Hilbert
solutions, high-order hybrid-pattern solitons are found,
which describe the nonlinear interaction between sev-
eral types of solitons.
B. Yang · Y. C h e n
Shanghai Key Laboratory of Trustworthy Computing, East
China Normal University, Shanghai 200062, China
Y. C h e n (
B
)
Department of Physics, Zhejiang Normal University,
Jinhua 321004, China
e-mail: profchenyong@163.com
Keywords Nonlocal nonlinear Schrödinger equa-
tions · High-order soliton · Riemann–Hilbert method
1 Introduction
The integrable nonlinear wave equations and soliton
theory have been studied for many years [1–5]. They
are significant subjects in many branches of nonlin-
ear science. For the integrable nonlinear models, most
of them are local equations, i.e., the solutions evolu-
tion depends only on the local solution value with its
local space and time derivatives. Recently, a number
of nonlocal integrable equations were found and trig-
gered renewed interest in integrable systems. The first
such nonlocal equation was the PT -symmetric nonlin-
ear Schrödinger (NLS) equation [6,7]:
iq
t
(x, t) = q
xx
(x, t) + 2q
2
(x, t)q
∗
(− x , t ), (1)
where asterisk * represents complex conjugation. For
this equation, the evolution of the solution at location
x depends both on the local position x and the distant
position − x. This implies that the states of the solu-
tion at distant opposite locations are directly related,
reminiscent of quantum entanglement in pairs of parti-
cles. Mathematically, this nonlocal integrable equation
is interesting and distinctly different from local equa-
tions. In the view of potential applications, this equation
was linked to an unconventional system of magnetics
[8]. Moreover, since Eq. (1) is parity-time (PT )sym-
metric, it is related to the concept of PT -symmetry,
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