Multilinear estimates and well‐posedness for the vortex filament fourth‐order Schrödinger equations

This paper is concerned with the initial value problem for the fourth‐order nonlinear Schrödinger type equation related to the theory of vortex filament. By deriving a fundamental estimate on dyadic blocks for the fourth‐order Schrödinger through the [k,Z]‐multiplier norm method. we establish multilinear estimates for this nonlinear fourth‐order Schrödinger type equation. The local well‐posedness for initial data in with s > 1 ∕ 2 is implied by the multilinear estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.     

The variational approximation for 2‐dimension solitons at cubic‐quintic nonlinearity in quasiperiodic potentials

We apply the variational approximation to study the dynamics of solitary waves of the nonlinear Schrödinger equation with compensative cubic‐quintic nonlinearity for asymmetric 2‐dimension setup. Such an approach allows to study the behavior of the solitons trapped in quasisymmetric potentials without an axial symmetry. Our analytical consideration allows finding the soliton profiles that are stable in a quasisymmetric geometry. We show that small perturbations of such states lead to generation of the oscillatory‐bounded solutions having 2 independent eigenfrequencies relating to the quintic nonlinear parameter. The behavior of solutions with large amplitudes is studied numerically. The resonant case when the frequency of the time variations (time managed) potential is near of the eigenfrequencies is studied too. In a resonant situation, the solitons acquire a weak time decay.     

On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic–quintic nonlinear Schrödinger equation with a periodic potential.     

Global Well-Posedness for Cubic NLS with Nonlinear Damping

We study the Cauchy problem for the cubic nonlinear Schrödinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we treat the energy-critical case of a quintic dissipation in three space dimensions.     

The sharp Strichartz and Sobolev–Strichartz inequalities for the fourth‐order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non‐endpoint Strichartz inequalities for the fourth‐order Schrödinger equation with initial data in the L²( R ^d) space in all dimensions, and then we obtain a maximizer also for the non‐endpoint Sobolev–Strichartz inequality for the fourth‐order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

Remark on energy‐critical NLS in 5D

We reprove global well‐posedness and scattering for the defocusing energy‐critical nonlinear Schrödinger equation in five dimensions. Inspired by the recent work of Killip and Visan, we adapt the Dodson's strategy ‘long‐time Strichartz estimate’ used in the work on mass‐critical nonlinear Schrödinger equation sets. Copyright © 2015 John Wiley & Sons, Ltd.     

Exact traveling wave solutions to the fourth‐order dispersive nonlinear Schrödinger equation with dual‐power law nonlinearity

In this paper, we investigate exact traveling wave solutions of the fourth‐order nonlinear Schrödinger equation with dual‐power law nonlinearity through Kudryashov method and (G'/G)‐expansion method. We obtain miscellaneous traveling waves including kink, antikink, and breather solutions. These solutions may be useful in the explanation and understanding of physical behavior of the wave propagation in a highly dispersive optical medium. Copyright © 2015 John Wiley & Sons, Ltd.     

Correspondence between the NLS equation for optical fibers and a class of integrable NLS equations

The propagation of the optical field complex envelope in a single‐mode fiber is governed by a one‐dimensional cubic nonlinear Schrödinger equation with a loss term. We present a result about L²‐closeness of the solutions of the aforementioned equation and of a one‐dimensional nonlinear Schrödinger equation that is Painlevé integrable. Copyright © 2014 John Wiley & Sons, Ltd.     

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Czechoslovak Mathematical Journal - We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the...     

Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation

We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic Solution of the Weakly Nonlinear Schrödinger Equation with Variable Coefficients

A perturbation method based on Fourier analysis and multiple scales is introduced for solving weakly nonlinear, dispersive wave propagation problems with Fourier-transformable initial conditions. Asymptotic solutions are derived for the weakly nonlinear cubic Schrödinger equation with variable coefficients, and verified by comparison with numerical solutions. In the special case of constant coefficients, the asymptotic solution agrees to leading order with previously derived results in the literature; in general, this is not true to higher orders. Therefore previous asymptotic results for the strongly nonlinear Schrödinger equation can be valid only for restricted initial conditions.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations

By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Schrödinger equations (DNLS). These transformations are shown to lead to two solution formulas for DNLS in terms of determinants. As applications, several different types of high‐order solutions are calculated for this equation.     

On Scattering for the Quintic Defocusing Nonlinear Schrödinger Equation on R × T2

We consider the problem of large‐data scattering for the quintic nonlinear Schrödinger equation on R × T ². This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a “large‐scale profile”) that controls the asymptotic behavior of the solutions. © 2014 Wiley Periodicals, Inc.     

On Multi‐component Ermakov Systems in a Two‐Layer Fluid: Integrable Hamiltonian Structures and Exact Vortex Solutions

By introducing an elliptic vortex ansatz, the 2+1‐dimensional two‐layer fluid system is reduced to a finite‐dimensional nonlinear dynamical system. Time‐modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen‐Ermakov‐Pinney equation.     

Cubic Nonlinear Schrödinger Equation on Three Dimensional Balls with Radial Data

We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of X ^{s, b} spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrödinger flow with particular initial data.     

Multidomain spectral method for Schrödinger equations

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.