Mohammed Elhadj Khelifa, Al Sakkaf L, Al Khawaja U, Boudjemâa Abdelâali
Department of Physics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria.
Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Chlef, Algeria.
Phys Rev E. 2020 Apr;101(4-1):042221. doi: 10.1103/PhysRevE.101.042221.
We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that this solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain an exact solution to the nonlocal NLSE using the same method and seed solution. The solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.
我们使用达布变换方法得到了局部非线性薛定谔方程（NLSE）的精确解。该解描述了双孤子分子的轮廓和动力学。利用代数衰减的种子解，我们得到了具有发散峰值的双孤子解，我们将其称为奇异孤子分子。我们发现这个解具有有限的束缚能。我们计算了两个孤子之间的相互作用力和势，结果表明它们是分子类型的。两个孤子之间键的稳健性也得到了验证。此外，我们使用相同的方法和种子解得到了非局部NLSE的精确解。在这种情况下，该解对应于一个孤子、平坦背景上的呼吸子孤子以及具有线性斜坡背景上的呼吸子孤子的弹性碰撞。最后，我们考虑一个时间而非空间非局部的NLSE。尽管我们没有找到该方程的拉克斯对，但我们推导出了三个精确解。
