By constructing the general six-parameter bright two-soliton solution of the integrable coupled nonlinear Schrödinger equation (Manakov model) using the Hirota method, we find that the solitons exhibit certain novel inelastic collision properties, which have not been observed in any other (1+1) dimensional soliton system so far. In particular, we identify the exciting possibility of switching solitons between modes by changing the phase. However, the standard elastic collision property of solitons is regained with specific choices of parameters.
We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schrödinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general N -coupled nonlinear Schrödinger equations (N -CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyse the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partial coherent solitons (PCS) discussed in the literature are special cases of the higher order bright soliton solutions of the N -CNLS equations.PACS numbers: 42.81. Dp, 42.65.Tg In recent years the concept of soliton has been receiving considerable attention in optical communications since soliton is capable of propagating over long distances without change of shape and with negligible attenuation [1][2][3]. It has been found that soliton propagation through optical fiber arrays is governed by a set of equations related to the CNLS equations [1,2],where q j is the envelope in the jth core, z and t represent the normalized distance along the fiber and the retarded time, respectively. Here 2µ gives the strength of the nonlinearity. Eq. (1) reduces to the standard envelope soliton possessing integrable nonlinear Schrödinger equation for N = 1. For N = 2, the above Eq. (1) governs the integrable Manakov system [4] and recently for this case the exact two-soliton solution has been obtained and novel shape changing inelastic collision property has been brought out [5]. However, the results are scarce for N ≥ 3, even though the underlying systems are of considerable physical interest. For example, in addition to optical communication, in the context of biophysics the case N = 3 can be used to study the launching and propagation of solitons along the three spines of an alpha-helix in protein [6]. Similarly the CNLS Eq.(1) and its generalizations for N ≥3 are of physical relevance in the theory of soliton wavelength division multiplexing [7], multi-channel bit-parallel-wavelength optical fiber networks [8] and so on. In particular, for arbitrary N , Eq.(1) governs the propagation of N -self trapped mutually incoherent wavepackets in Kerr-like photorefractive media [9] in which q j is the jth component of the beam, z and t represents the normalized coordinates along the direction of propagation and the transverse coordinate, respectively, and N p=1 |q p | 2 represents the change in the refractive index profile created by all incoherent components of the light beam [9] when the medium response is slow.The parameter µ = k 2 0 n 2 /2, where n 2 is the nonlinear Kerr coefficient and k 0 is the free space wave vector.In this letter, we report the exact bright one and two soliton solutions, first for the N = 3 case and then for the arbitrary N case, where the procedure can be extended in principle to higher order soliton solutions, using the Hirota bilinearization method. In particular, ...
In this note we present a remarkable nonlinear system which has the property that all its bounded periodic motions are simple harmonic. The system is a particle obeying the highly nonlinear equation of motionwhence the canonical momentum p and the Hamiltonian II may be obtained:The Lagrangian (2) is the single particle analogue of the Lagrangian densityof a relativistic scalar field. Field systems with this kind of Lagrangian are of wide current interest [1,2] in the context of elementary particle theory, and have in fact been considered especially in the context of chiral Lagrangian theories of pion interactions. Here we limit ourselves to obtaining the solutions of Eq. (1), which immediately lead to the plane wave (c-number) solutions of the field equation of (5) where i(y) = f f(y) dy and the constant of integration has been written as -(«/X) for future convenience.
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