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Register a new userAnalytic iteration procedure for solitons and traveling wavefronts with sources
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A method is presented for calculating solutions to differential equations analytically for a variety of problems in physics. An iteration procedure based on the recently proposed BLUES (Beyond Linear Use of Equation Superposition) function method is shown to converge for nonlinear ordinary differential equations. Case studies are presented for solitary wave solutions of the Camassa-Holm equation and for traveling wavefront solutions of the Burgers equation, with source terms. The convergence of the analytical approximations towards the numerically exact solution is exponentially rapid. In practice, the zeroth-order approximation (a simple convolution) is already useful and the first-order approximation is already accurate while still easy to calculate. The type of nonlinearity can be chosen rather freely, which makes the method generally applicable.
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We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrodinger (NLS) equations with generalized parity-time (PT)-symmetric Scarff II potentials. Particularly, a PT-symmetric k-wavenumber Scarff II potential and a multi-well Scarff II potential are considered, respectively. For the k-wavenumber Scarff II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT-symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multi-well Scarff II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wavenumber and multi-well Scarff II potentials is analyzed in details using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT-symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.
Based on conservation laws as one of the important integrable properties of nonlinear physical models, we design a modified physics-informed neural network method based on the conservation law constraint. From a global perspective, this method imposes physical constraints on the solution of nonlinear physical models by introducing the conservation law into the mean square error of the loss function to train the neural network. Using this method, we mainly study the standard nonlinear Schrodinger equation and predict various data-driven optical soliton solutions, including one-soliton, soliton molecules, two-soliton interaction, and rogue wave. In addition, based on various exact solutions, we use the modified physics-informed neural network method based on the conservation law constraint to predict the dispersion and nonlinear coefficients of the standard nonlinear Schrodinger equation. Compared with the traditional physics-informed neural network method, the modified method can significantly improve the calculation accuracy.
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.
In the present work, we explore the potential of spin-orbit (SO) coupled Bose-Einstein condensates to support multi-component solitonic states in the form of dark-bright (DB) solitons. In the case where Raman linear coupling between components is absent, we use a multiscale expansion method to reduce the model to the integrable Melnikov system. The soliton solutions of the latter allow us to reconstruct approximate traveling DB solitons for the reduced SO coupled system. For small values of the formal perturbation parameter, the resulting waveforms propagate undistorted, while for large values thereof, they shed some dispersive radiation, and subsequently distill into a robust propagating structure. After quantifying the relevant radiation effect, we also study the dynamics of DB solitons in a parabolic trap, exploring how their oscillation frequency varies as a function of the bright component mass and the Raman laser wavenumber.
The iteration sequence based on the BLUES (Beyond Linear Use of Equation Superposition) function method for calculating analytic approximants to solutions of nonlinear ordinary differential equations with sources is elaborated upon. Diverse problems in physics are studied and approximate analytic solutions are found. We first treat a damped driven nonlinear oscillator and show that the method can correctly reproduce oscillatory behaviour. Next, a fractional differential equation describing heat transfer in a semi-infinite rod with Stefan-Boltzmann cooling is handled. In this case, a detailed comparison is made with the Adomian decomposition method, the outcome of which is favourable for the BLUES method. As a final problem, the Fisher equation from population biology is dealt with. For all cases, it is shown that the solutions converge exponentially fast to the numerically exact solution, either globally or, for the Fisher problem, locally.
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