The defocusing nonlinear Schrödinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in the defocusing NLS equation with real-valued time-dependent and time-independent potentials such that the stable new RWs and W-shaped solitons are found, respectively. Moreover, the interactions of two or three RWs are explored such that the RWs with higher amplitudes are generated in the defocusing NLS equation with real-valued time-dependent potentials. Finally, we study the defocusing NLS equation with complex PT -symmetric potentials such that some RWs and W-shaped solitons are also found. These novel results will be useful to design the related physical experiments to generate the RW phenomena and W-shaped solitons in the case of defocusing nonlinear interactions, and to apply them in the related fields of nonlinear or even linear sciences.
We present exact rational solution for a modified nonlinear Schr$\ddot{o}$dinger equation that takes into account quintic nonlinearity and nonlinear dispersion corrections to the cubic nonlinearity, which could be used to describe rogue wave in nonlinear fibers. We find the rogue wave with these higher order effects has identical shape with the well-known one in nonlinear Schr$\ddot{o}$dinger equation. However, the quintic nonlinear term and nonlinear dispersion effect affect the velocity of rogue wave, and the evolution of its phase.
It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schr{ö}dinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schr{ö}dinger equation. The additional compression points are generated in pairs forming a comb-like structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The…
In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlinear Schrodinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrodinger system with PT-symmetric nonlinearity also admits Peregrine Soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we o…
This article addresses the question of general rogue-wave solutions in the nonlocal parity-time-symmetric nonlinear Schrodinger equation. By generalizing the previous bilinear Kadomtsev-Petviashvili reduction method, large classes of rogue waves are derived as Gram determinants with Schur polynomial elements. It is shown that these rogue waves contain previously reported ones as special cases. More importantly, they contain many new rogue wave families. It is conjectured that the rogue waves derived in this article are all rogue-wave solutions in the parity-time-symmetric nonlinear Schrodinger…
We investigate rogue-wave solutions in a three-component coupled nonlinear Schrodinger equation. With certain requirements on the backgrounds of components, we construct a multi-rogue-wave solution that exhibits a structure like a four-petaled flower in temporal-spatial distribution, in contrast to the eye-shaped structure in one-component or two-component systems. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.
We construct vector rogue wave solutions of the two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients, namely diffraction, nonlinearity and gain parameters through similarity transformation technique. We transform the two-dimensional two coupled variable coefficients nonlinear Schrödinger equations into Manakov equation with a constraint that connects diffraction and gain parameters with nonlinearity parameter. We investigate the characteristics of the constructed vector rogue wave solutions with four different forms of diffraction parameters. We report some…
Rogue wave patterns in the nonlinear Schrödinger equation are analytically studied. It is shown that when an internal parameter in the rogue waves (which controls the shape of initial weak perturbations to the uniform background) is large, these waves would exhibit clear geometric structures, which are formed by Peregrine waves in shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controll…
We explore the form of rogue wave solutions in a select set of case examples of nonlinear Schrödinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose-Einstein condensates in different experimentally relevant settings. For these models, we identify exact rogue wave solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence …
We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schrödinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for N-th order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.
Modulation instability, rogue wave and spectral analysis are investigated for the nonlinear Schrodinger equation with the higher-order terms. The modulation instability distribution characteristics from the sixth-order to the eighth-order nonlinear Schrodinger equations are studied. Higher-order dispersion terms are closely related to the distribution of modulation stability regime, and n-order dispersion term corresponds to $n-2$ modulation stability curves in the modulation instability band. Based on the generalized Darboux transformation method, the higher-order rational solutions are const…
We study on dynamics of high-order rogue wave in two-component coupled nonlinear Schrödinger equations. We find four fundamental rogue waves can emerge for second-order vector RW in the coupled system, in contrast to the high-order ones in single component systems. The distribution shape can be quadrilateral, triangle, and line structures through varying the proper initial excitations given by the exact analytical solutions. Moreover, six fundamental rogue wave can emerge on the distribution for second-order vector rogue wave, which is similar to the scalar third-order ones. The distribution p…
We present exact solutions for rogue waves arising on the background of periodic waves in the focusing nonlinear Schrodinger equation. The exact solutions are obtained by characterizing the Lax spectrum related to the periodic waves and by using the one-fold Darboux transformation. The magnification factor of the rogue waves is computed in the closed analytical form. We relate the rogue wave solutions to the modulation instability of the periodic wave background.
We construct breather and rogue wave solutions of a variable coefficient nonlinear Schrödinger equation with an external linear potential. This generalized model describes the nonlinear wave propagation in an inhomogeneous plasma/medium. We derive several localized solutions including Ma breather, Akhmediev breather, two-breather and rogue wave solutions of this model and show how the inhomogeneity of space modifies the shape and orientation of these localized structures. We also depict the trajectories of the inhomogeneous rogue wave. Our results may be useful for controlling plasmonic energy…
We study on dynamics of high-order rogue wave in two-component coupled nonlinear Schrödinger equations. Based on the generalized Darboux transformation and formal series method, we obtain the high-order rogue wave solution without the special limitation on the wave vectors. As an application, we exhibit the first, second-order rogue wave solution and the superposition of them by computer plotting. We find the distribution patterns for vector rogue waves are much more abundant than the ones for scalar rogue waves, and also different from the ones obtained with the constrain conditions on backgr…
Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only $n(n+1)$, but also $n(n-1)+1$ and $n^2$, where $n$ is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time, or …
Breathers and rogue waves of special coupled nonlinear Schrödinger systems (the Manakov equations) are studied analytically. These systems model the orthogonal polarization modes in an optical fiber with randomly varying birefringence. Unlike the situation in a waveguide with zero birefringence, rogue waves can occur in these Manakov systems with dispersion and nonlinearity of opposite signs, provided that a group velocity mismatch is present. The criterion for the existence of rogue waves correlates exactly with the onset of modulation instability. Theoretically the Hirota bilinear transform …
We construct Darboux transformation of a coupled generalized nonlinear Schrödinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these solutions. We show that the rogue wave solution can be found only when the four wave mixing parameter becomes real. We also investigate the modulation instability of the steady state solution of CGNLS system and demonstrate that it can occur only when the four wave mixing parameter becomes real. Our results give an evidence for the connection between the occurrence of…
The perturbation theory based on the Riemann-Hilbert problem is developed for the modified nonlinear Schr{ö}dinger equation which describes the propagation of femtosecond optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The linear perturbation and the Raman gain are considered as examples.
The double-periodic solutions of the focusing nonlinear Schrodinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on their background. Magnification of the rogue waves is studied numerically.
We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schrödinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |Γt| \ll 1, with Γ the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.
In this paper, a simple and constructive method is presented to find the generalized perturbation (n,M)-fold Darboux transformations (DTs) of the modified nonlinear Schrodinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1,N-1)-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc. and may be useful to study …
General rogue waves are derived for the generalized derivative nonlinear Schrodinger (GDNLS) equations by a bilinear Kadomtsev-Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a n…
We construct rogue wave solutions of a fifth-order nonlinear Schrödinger equation on the Jacobian elliptic function background. By combining Darboux transformation and the nonlinearization of spectral problem, we generate rogue wave solution on two different periodic wave backgrounds. We analyze the obtained solutions for different values of system parameter and point out certain novel features of our results. We also compute instability growth rate of both $dn$ and $cn$ periodic background waves for the considered system through spectral stability problem. We show that instability growth rate…
We consider a coherently coupled nonlinear Schrödinger equation with modulated self-phase modulation, cross-phase modulation, and four-wave mixing nonlinearities and varying refractive index in anisotropic graded index nonlinear medium. By identifying an appropriate similarity transformation, we obtain a general localized wave solution and investigate their dynamics with a proper set of modulated nonlinearities. In particular, our study reveals different manifestations of localized waves such as stable solitons, Akhmediev breathers, Ma breathers, and rogue waves of bright, bright-dark, and dar…
