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 $N \geq 3$ 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 $N = 3$ and for cubic nonlinearity.Comment: 23 page

Rogue periodic waves of the focusing NLS equation

Rogue waves on the periodic background are considered for the nonlinear Schrodinger (NLS) equation in the focusing case. The two periodic wave solutions are expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue waves on the periodic background are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions labeled as rogue periodic waves generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the wave amplitude (the elliptic modulus). Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.Comment: 20 pages, 3 figure

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 $H^2(\mathbb{R}) \cap H^{1,1}(\mathbb{R})$ 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

Rogue periodic waves of the mKdV equation

Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation are considered in the focusing case. By using one-fold and two-fold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation expressed by the Jacobian elliptic functions dn and cn respectively. The rogue dn-periodic wave describes propagation of an algebraically decaying soliton over the dn-periodic wave, the latter wave is modulationally stable with respect to long-wave perturbations. The rogue cn-periodic wave represents the outcome of the modulation instability of the cn-periodic wave with respect to long-wave perturbations and serves for the same purpose as the rogue wave of the nonlinear Schrodinger equation (NLS), where it is expressed by the rational function. We compute the magnification factor for the cn-periodic wave of the mKdV equation and show that it remains the same as in the small-amplitude NLS limit for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the AKNS spectral problem associated with the dn- and cn-periodic waves of the mKdV equation.Comment: 24 pages, 3 figure

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 $N$ 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 $N$ under the classical Kirchhoff boundary conditions and (ii) the star graph with one incoming edge and $N-1$ 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 $N \geq 4$ 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 $N-1$ outgoing edges are spectrally stable, whereas the shifted states with non-monotone profiles in the $N-1$ 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