146 research outputs found
On the linearized log-KdV equation
The logarithmic KdV (log-KdV) equation admits global solutions in an energy
space and exhibits Gaussian solitary waves. Orbital stability of Gaussian
solitary waves is known to be an open problem. We address properties of
solutions to the linearized log-KdV equation at the Gaussian solitary waves. By
using the decomposition of solutions in the energy space in terms of Hermite
functions, we show that the time evolution is related to a Jacobi difference
operator with a limit circle at infinity. This exact reduction allows us to
characterize both spectral and linear orbital stability of solitary waves. We
also introduce a convolution representation of solutions to the log-KdV
equation with the Gaussian weight and show that the time evolution in such a
weighted space is dissipative with the exponential rate of decay.Comment: 18 page
Normal form for transverse instability of the line soliton with a nearly critical speed of propagation
There exists a critical speed of propagation of the line solitons in the
Zakharov-Kuznetsov (ZK) equation such that small transversely periodic
perturbations are unstable for line solitons with larger-than-critical speeds
and orbitally stable for those with smaller-than-critical speeds. The normal
form for transverse instability of the line soliton with a nearly critical
speed of propagation is derived by means of symplectic projections and
near-identity transformations. Justification of this normal form is provided
with the energy method. The normal form predicts a transformation of the
unstable line solitons with larger-than-critical speeds to the orbitally stable
transversely modulated solitary waves.Comment: 20 page
Spectral instability of the peaked periodic wave in the reduced Ostrovsky equation
We show that the peaked periodic traveling wave of the reduced Ostrovsky
equations with quadratic and cubic nonlinearity is spectrally unstable in the
space of square integrable periodic functions with zero mean and the same
period. The main novelty is that we discover a new instability phenomenon: the
instability of the peaked periodic waves is induced by spectrum of a linearized
operator which completely covers a closed vertical strip of the complex plane.Comment: 15 pages; 1 figur
Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background
We address the most general periodic travelling wave of the modified
Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian
elliptic functions. By applying an algebraic method which relates the periodic
travelling waves and the squared periodic eigenfunctions of the Lax operators,
we characterize explicitly the location of eigenvalues in the periodic spectral
problem away from the imaginary axis. We show that Darboux transformations with
the periodic eigenfunctions remain in the class of the same periodic travelling
waves of the mKdV equation. In a general setting, there are exactly three
symmetric pairs of eigenvalues away from the imaginary axis, and we give a new
representation of the second non-periodic solution to the Lax equations for the
same eigenvalues. We show that Darboux transformations with the non-periodic
solutions to the Lax equations produce rogue waves on the periodic background,
which are either brought from infinity by propagating algebraic solitons or
formed in a finite region of the time-space plane.Comment: 40 pages, 8 figure
Convergence of Petviashvili's method near periodic waves in the fractional Korteweg-de Vries equation
Petviashvili's method has been successfully used for approximating of
solitary waves in nonlinear evolution equations. It was discovered empirically
that the method may fail for approximating of periodic waves. We consider the
case study of the fractional Korteweg-de Vries equation and explain divergence
of Petviashvili's method from unstable eigenvalues of the generalized
eigenvalue problem. We also show that a simple modification of the iterative
method after the mean value shift results in the unconditional convergence of
Petviashvili's method. The results are illustrated numerically for the
classical Korteweg-de Vries and Benjamin-Ono equations.Comment: 30 pages; 13 figure
Nonlinear Instability of Half-Solitons on Star Graphs
We consider a half-soliton stationary state of the nonlinear Schrodinger
equation with the power nonlinearity on a star graph consisting of N edges and
a single vertex. For the subcritical power nonlinearity, the half-soliton state
is a degenerate critical point of the action functional under the mass
constraint such that the second variation is nonnegative. By using normal
forms, we prove that the degenerate critical point is a nonlinear saddle point,
for which the small perturbations to the half-soliton state grow slowly in time
resulting in the nonlinear instability of the half-soliton state. The result
holds for any and arbitrary subcritical power nonlinearity. It gives
a precise dynamical characterization of the previous result of Adami {\em et
al.}, where the half-soliton state was shown to be a saddle point of the action
functional under the mass constraint for and for cubic nonlinearity.Comment: 23 page
Existence of global solutions to the derivative NLS equation with the inverse scattering transform method
We address existence of global solutions to the derivative nonlinear
Schr\"{o}dinger (DNLS) equation without the small-norm assumption. By using the
inverse scattering transform method without eigenvalues and resonances, we
construct a unique global solution in which is also Lipschitz continuous with respect to the
initial data. Compared to the existing literature on the spectral problem for
the DNLS equation, the corresponding Riemann--Hilbert problem is defined in the
complex plane with the jump on the real line
Stability of multi-solitons in the cubic NLS equation
We address stability of multi-solitons in the cubic NLS (nonlinear
Schr\"{o}dinger) equation on the line. By using the dressing transformation and
the inverse scattering transform methods, we obtain the orbital stability of
multi-solitons in the space when the initial data is in a
weighted space
Long-time stability of breathers in Hamiltonian -symmetric lattices
We consider the Hamiltonian version of a -symmetric lattice that
describes dynamics of coupled pendula under a resonant periodic force. Using
the asymptotic limit of a weak coupling between the pendula, we prove the
nonlinear long-time stability of breathers (time-periodic solutions localized
in the lattice) by using the Lyapunov method. Breathers are saddle points of
the extended energy function, which are located between the continuous bands of
positive and negative energy. Nevertheless, we construct an approximate
Lyapunov function and estimate its evolution on a long but finite time
interval. The nonlinear stability analysis becomes possible for the -symmetric lattice only because of the existence of a Hamiltonian structure
Spectral stability of shifted states on star graphs
We consider the nonlinear Schr\"{o}dinger (NLS) equation with the subcritical
power nonlinearity on a star graph consisting of edges and a single vertex
under generalized Kirchhoff boundary conditions. The stationary NLS equation
may admit a family of solitary waves parameterized by a translational
parameter, which we call the shifted states. The two main examples include (i)
the star graph with even under the classical Kirchhoff boundary conditions
and (ii) the star graph with one incoming edge and outgoing edges under a
single constraint on coefficients of the generalized Kirchhoff boundary
conditions. We obtain the general counting results on the Morse index of the
shifted states and apply them to the two examples. In the case of (i), we prove
that the shifted states with even are saddle points of the action
functional which are spectrally unstable under the NLS flow. In the case of
(ii), we prove that the shifted states with the monotone profiles in the
outgoing edges are spectrally stable, whereas the shifted states with
non-monotone profiles in the outgoing edges are spectrally unstable, the
two families intersect at the half-soliton states which are spectrally stable
but nonlinearly unstable. Since the NLS equation on a star graph with shifted
states can be reduced to the homogeneous NLS equation on a line, the spectral
instability of shifted states is due to the perturbations breaking this
reduction. We give a simple argument suggesting that the spectrally stable
shifted states are nonlinear unstable under the NLS flow due to the
perturbations breaking the reduction to the NLS equation on a line.Comment: 22 pages, 3 figure
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