K. Porsezian

Generating mechanism for higher-order rogue waves

We introduce a mechanism for generating higher-order rogue waves (HRWs) of the nonlinear Schrödinger (NLS) equation: the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue λ(0) creates an order-n HRW. By adjusting the relative phase of the breathers in the interacting area, it is possible to obtain different types of HRWs. The value λ(0) is a zero point of an eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order-n HRW.

On the integrability aspects of the one‐dimensional classical continuum isotropic biquadratic Heisenberg spin chain

The integrability aspects of a classical one‐dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher‐order generalized nonlinear Schrodinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower‐order continuum isotropic spin chain, is identified by carrying out a Painleve singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.

Effect of discreteness on the continuum limit of the Heisenberg spin chain

We discuss the effect of discreteness of the lattice on the classical continuum limit of the isotropic Heisenberg ferromagnetic spin chain, including biquadratic interaction, in the next order. We derive the equivalent higher order nonlinear Schrodinger equation by identifying the underlying geometry of the system and investigate its non-integrability nature. A multiple-scaling method is applied to obtain the perturbed soliton solution.

Rogue waves of the Hirota and the Maxwell-Bloch equations

In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch (H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the n-fold Darboux transformation of this system. This n-fold Darboux transformation implies the determinant representation of nth solutions of (E([n]),p([n]),η([n])) generated from the known solution of (E,p,η). The determinant representation of (E([n]),p([n]),η([n])) provides soliton solutions, positon solutions, and breather solutions (both bright and dark breathers) of the H-MB system. From the breather solutions, we also construct a bright and dark rogue wave solution for the H-MB system, which is currently one of the hottest topics in mathematics and physics. Surprisingly, the rogue wave solution for p and η has two peaks because of the order of the numerator and denominator of them. Meanwhile, after fixing the time and spatial parameters and changing two other unknown parameters α and β, we generate a rogue wave shape.

Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation.

In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.

On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions

The dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n‐) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrodinger equation. A Painleve singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n‐) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the form f(r)=e1r−2(n−1)+e2r−(n−2). This is confirmed by obtaining the associated Lax pair, Backlund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one‐dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher‐dimensional inhomogeneous spherically symmetric spin models can be formally mapped.

Soliton Interaction Under Soliton Dispersion Management

The concept of soliton dispersion management pertaining to the effect of varying dispersion with external harmonic oscillator potential for chirped solitons have been studied in detail with emphasis on the various aspects of dispersion management of solitons. The interaction scenarios pertaining to in-phase- and off-phase injection with equal and unequal amplitudes are dealt elaborately. Suppression of interaction forces between the dispersion managed solitons, optical soliton pulse amplification and robustness of bound soliton states to collisions, are some of the main results of this article.

New Types of Rogue Wave in an Erbium-Doped Fibre System

We report a novel and new types of rogue optical wave propagation in an erbium-doped fibre system governed by the nonlinear Schrodinger and the Maxwell–Bloch equation. The breather solutions of the three fields, namely field envelop, polarization and population inversion, are used to generate the rogue waves. For the first time, we report bright and, in particular, dark rogue waves in a coupled nonlinear optical systems. The distinction between bright and dark rogue waves are discussed in detail through figures. The rogue wave formation in our model can also be connected to the generation of supercontinuum generation in resonant optical fibre.

Few-cycle optical rogue waves: Complex modified Korteweg-de Vries equation

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.

On the dynamics of the radially symmetric Heisenberg ferromagnetic spin system

By considering the geometrical equivalence of the radially symmetric Heisenberg ferromagnetic spin system in n‐arbitrary spatial dimensions and the generalized nonlinear Schrodinger equation (GNLSE) with radial symmetry, it is shown that they possess the Painleve property only for the (n=2) circularly (planar radially) symmetric case. For the circularly symmetric case, suitable (2×2) matrix eigenvalue equations are constructed, involving nonisospectral flows and their gauge equivalence is shown. The connection with inhomogeneous systems and, in particular, the linearly x‐dependent system is pointed out. Appropriate Backlund transformations (BT) and explicit soliton solutions for both the spin systems and the GNLSEs are also derived.