We introduce a mechanism for generating higher order rogue waves (HRWs) of the nonlinear Schr\"odinger(NLS) equation: progressive fusion and fission $n$ degenerate breathers associated with critical eigenvalue $\lambda_0$, creates an HRW. By adjusting relative phase at interacting area, it is possible to obtain different types HRWs. The value $\lambda_0$ zero point eigenfunction Lax pair NLS equation corresponds limit period breather tending infinity. employing this we prove two conjectures...

The determinant representation of the n-fold Darboux transformation Hirota equation is given. Based on our analysis, 1-soliton, 2-soliton, and breathers solutions are given explicitly. Further, first order rogue wave by a Taylor expansion breather solutions. In particular, explicit formula has several parameters, which more general than earlier reported results thus provides systematic way to tune experimentally waves choosing different values for them.

The n-fold Darboux transformation (DT) is a 2\times2 matrix for the Kaup-Newell (KN) system. In this paper,each element of expressed by ratio $(n+1)\times (n+1)$ determinant and $n\times n$ eigenfunctions. Using these formulae, expressions $q^{[n]}$ $r^{[n]}$ in KN system are generated DT. Further, under reduction condition, rogue wave,rational traveling solution, dark soliton, bright breather periodic solution derivative nonlinear Schr\"odinger(DNLS) equation given explicitly different seed...

Abstract In this paper, the partially party‐time ( ) symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called ‐nonlocal DS equations, while a fully DSII equation equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these are derived by employing bilinear method. For usual (2 + 1)‐dimensional breathers periodic in direction localized y direction. Nonsingular rational lumps, composed breathers, line waves. equation, both...

In this paper, using the Darboux transformation, we demonstrate generation of first order breather and higher-order rogue waves from a generalized nonlinear Schr\"odinger equation with several effects representing femtosecond pulse propagation through silica fiber. The same evolution can also describes soliton-type excitations in classical Heisenberg spin chain. Such solutions have parameter $\gamma_1$, denoting strength effects. From numerical plots rational solutions, compression produced...

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using Lax pair, construct generalized Darboux transformation and systematically generate first-, second-, third-order rogue wave solutions analyze nature evolution higher-order waves in detail. Based on detailed numerical analytical investigations, classify with respect to their intrinsic structure, namely, fundamental pattern, triangular ring pattern. We also present...

The Fokas-Lenells (FL) equation arises as a model eqution which describes for nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order (in Schr\"odinger (NLS) results). Here we present an explicit analytical representation rogue waves of FL equation. This is constructed deriving appropriate Darboux transformation (DT) and utilizing Taylor series expansion associated breather solution. when certain higher-order effects are considered, given.

In this paper, we derive a Darboux transformation of the Hirota and Maxwell-Bloch(H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fibre further generalize it to matrix form $n$-fold system. This implies determinant representation $n$-th new solutions $(E^{[n]},p^{[n]}, \eta^{[n]})$ generated from known solution $(E, p,\eta)$. The $(E^{[n]},p^{[n]} ,\eta^{[n]})$ provides soliton solutions, positon breather (both bright dark breathers) H-MB From also...

The rogue waves in a resonant erbium-doped fiber system governed by coupled of the nonlinear Schr\"odinger equation and Maxwell-Bloch (NLS-MB equations) are given explicitly Taylor series expansion about breather solutions normalized slowly varying amplitude complex field envelope $E$, polarization $p$, population inversion $\ensuremath{\eta}$. $n$-order three fields constructed using Darboux transformation (DT) assuming periodic seed solutions. Moreover, determinant forms with $n+3$ free...

Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is two-spatial dimensional analogue of the nonlinear Schrödinger (NLS) (Ablowitz and Musslimani 2013 Phys. Rev. Lett. 110 064105), involving self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves breathers are derived by employing bilinear method. The long wave limit these solutions yields two kinds fundamental rogue waves, namely,...

We construct an analytical and explicit representation of the Darboux transformation (DT) for Kundu–Eckhaus (KE) equation. Such solution n -fold DT T are given in terms determinants whose entries expressed by initial eigenfunctions ‘seed’ solutions. Furthermore, formulae higher order rogue wave (RW) solutions KE equation also obtained using Taylor expansion with use degenerate eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mi...

We construct higher order rogue wave solutions for the Gerdjikov-Ivanov equation explicitly in term of determinant expression.Dynamics both soliton and non-soliton is discussed.A family with distinct structures are presented, which new to equation.

In this paper, we construct a special kind of breather solution the nonlinear Schrödinger (NLS) equation, so-called breather-positon (b-positon for short), which can be obtained by taking limit λ_{j}→λ_{1} Lax pair eigenvalues in order-n periodic solution, is generated n-fold Darboux transformation from "seed" solution-plane wave. Further, an b-positon gives rogue wave under λ_{1}→λ_{0}. Here, λ_{0} eigenvalue NLS equation such that its period goes to infinity. Several analytical plots...

Abstract Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP‐I) equation are studied. The KP‐I describes evolution weakly nonlinear, dispersive waves with slow transverse variations. Lumps can only exist for KP when signs derivative weak dispersion in propagation direction different, that is, regime. Collisions “integrable” equations normally elastic, wave shapes preserved except possibly phase shifts. For resonant collisions, mathematically shift...

Resonant collisions of lumps with periodic solitons the Kadomtsev–Petviashvili I equation are investigated in detail. The usual lump is a stable weakly localized two-dimensional soliton, which keeps its shape and velocity course evolution from t → −∞ to +∞. However, would become time as instantons, result two types resonant spatially (quasi-1D) soliton chains. These partly fully collisions. In former case, does not exist at −∞, but it suddenly emerges chain, keeping amplitude constant +∞; or...

A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with superalgebras. Applications in the case superalgebra B(0,1) present super-AKNS soliton hierarchy and super-Dirac hierarchy.

The Gerdjikov-Ivanov (GI) system of q and r is defined by a quadratic polynomial spectral problem with 2 × matrix coefficients. Each element the n-fold Darboux transformation (DT) for this expressed ratio (n + 1) determinant n eigenfunctions, which implies representation q[n] r[n] generated from known solution r. By choosing some special eigenvalues eigenfunctions according to reduction conditions = −(r[n])*, provides new solutions GI equation. As examples, breather rogue wave are given...

Considering certain terms of the next asymptotic order beyond nonlinear Schrödinger equation, Fokas–Lenells (FL) equation governed by FL system arises as a model for pulse propagation in optical fibers. The expressions q[n] and r[n] are generated n-fold Darboux transformation (DT), each element matrix is 2 × matrix, expressed ratio (2n + 1) determinant 2n eigenfunctions. Further, Taylor series expansion about n-order breather solutions using DT assuming periodic seed under reduction can...