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 λ0 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 λ0 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.

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Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations Quantum chains with a Catalan tree pattern of conserved charges: The Δ=−1 XXZ model and the isotropic octonionic chain

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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 fibre. The same nonlinear evolution equation can also describes 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.

We propose a coupled system of the Hirota equation and the Maxwell-Bloch equations to describe the wave propagation in an erbium doped nonlinear fiber with higher order dispersion.The Painleve property of the same is analyzed and the coupled system is found to be integrable. The Lax pair is also constructed and the single-soliton solution is explicitly shown. The coupled system is found to allow soliton-type propagation. PACS numbers: 42.SO.Rh, 02.30.Jr, 42.65. -k, 42.81.DpThe extraordinary growth of communication technology was made possible only because of the discovery of microwaves. Every day the amount of intelligence to be communicated becomes more and more, so communication using light wave technology has been developed. Optical communication gives the necessary bandwidth, which can handle numerous channels. Optical fibers made of silica are used for guiding the light waves.Still the entire efficiency of optical communication is not utilized. This is because of the undesired, natural problems of the fibers. The two major problems are the dispersion and the dissipation due to the frequency dependence of the index of refraction and the optical losses, respectively.The dispersion makes the optical pulse spread temporally so that the energy may fall on the next bit slot, so that cross talk or error detection may occur. The optical losses cause the vanishing of the pulse because of absorption and scattering. This also causes error detection.The other important natural phenomenon of optical fibers is the nonlinear effect. When the intensity of the optical pulses crosses a certain threshold value then the fiber behaves nonlinearly. The most important effect is self-phase modulation (SPM) due to Kerr nonlinearity.Kerr nonlinearity is defined as the intensity-dependent refractive index. SPM produces spectral broadening. The effect of SPM is in opposition to that of dispersion in the anomalous dispersion region (the wavelength at which the blueshifted frequency component travels faster than the redshifted frequency).So with the proper selection of the parametric conditions, the effect of dispersion can be exactly balanced with the help of SPM. With this principle the pulse will travel through a very long distance without any change in shape -this effect is called the soliton [1,2].In 1980, Mollenauer, Stolen, and Gordon [3] observed experimental solitons for the first time in low-loss fibers.Later in 1986 [4], the same group reported that the experimentally observed solitons did not have the same properties as theoretical ones. This is. because of higher order effects like higher order dispersion, self-steepening, and stimulated inelastic scattering. All these effects will give some additional perturbation to the soliton system.It is evident that if one analyzes any type of soliton in fibers, all the above important additional effects have to be included. Here in this Letter we mainly concentrate on this problem of our proposed system.The other important nonlinear effect is the coherent interaction of the optical fi...

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.