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Register a new userTwo-component vector breather solution of the modified BBM equation
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G. T. Adamashvili
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This is a continuation of Ref.[1](arXiv:nlin.PS/2001.07758v1). In the present paper, we consider the solution to the modified Benjamin-Bona-Mahony equation $u_{ t} + C u_{z} + beta u_{zzt} + a u^{2} u_{z}=0$ using the generalized perturbation reduction method. The equation is transformed to the coupled nonlinear Schrodinger equations for auxiliary functions. Explicit analytical expression for the shape and parameters of the two-component vector breather oscillating with the sum and difference of frequencies and wavenumbers are obtained.
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Using the generalized perturbation reduction method the scalar nonlinear Schrodinger equation is transformed to the coupled nonlinear Schrodinger equations for auxiliary functions. A solution in the form of a two-component vector nonlinear pulse is obtained. The components of the pulse oscillate with the sum and difference of the frequencies and wave numbers. Explicit analytical expressions for the shape and parameters of the two-component nonlinear pulse are presented.
The generalized perturbative reduction method is used to find the two-component vector breather solution of the nonlinear Klein-Gordon equation. It is shown that the nonlinear pulse oscillates with the sum and difference of frequencies and wave numbers in the region of the carrier wave frequency and wave number. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are obtained. In the particular case, the vector breather coincides with the vector $0pi$ pulse of self-induced transparency.
New two-component vector breather solution of the modified Benjamin-Bona-Mahony (MBBM) equation is considered. Using the generalized perturbation reduction method the MBBM equation is reduced to the coupled nonlinear Schrodinger equations for auxiliary functions. Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented. The two-component vector breather and single-component scalar breather of the MBBM equation is compared.
In this paper, we provide the geometric formulation to the two-component Camassa-Holm equation (2-mCHE). We also study the relation between the 2-mCHE and the M-CV equation. We have shown that these equations arise from the invariant space curve flows in three-dimensional Euclidean geometry. Using this approach we have established the geometrical equivalence between the 2-mCHE and the M-CV equation. The gauge equivalence between these equations is also considered.
We derive the two-breather solution of the class I infinitely extended nonlinear Schrodinger equation (NLSE). We present a general form of this multi-parameter solution that includes infinitely many free parameters of the equation and free parameters of the two breather components. Particular cases of this solution include rogue wave triplets, and special cases of breather-to-soliton and rogue wave-to-soliton transformations. The presence of many parameters in the solution allows one to describe wave propagation problems with higher accuracy than with the use of the basic NLSE.
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