TY - JOUR

T1 - Singular soliton molecules of the nonlinear Schrödinger equation

AU - Mohammed Elhadj, Khelifa

AU - Al Sakkaf, L.

AU - Al Khawaja, U.

AU - Boudjemaâ, Abdelaâli

N1 - Funding Information:
K.M.E. and A.B. acknowledge support from the University of Chlef. L. Al Sakkaf and U. Al Khawaja acknowledge support from the UAE University through the Grants No. UAEU-UPAR(4) and No. UAEU-UPAR(6). K. M. Elhadj is grateful to UAEU for hosting him during part of the work on this paper.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/4

Y1 - 2020/4

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevE.101.042221

DO - 10.1103/PhysRevE.101.042221

M3 - Article

C2 - 32422845

AN - SCOPUS:85084612960

SN - 2470-0045

VL - 101

JO - Physical Review E

JF - Physical Review E

IS - 4

M1 - 042221

ER -