Research paperIntegrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities, PT-symmetric potentials, and space- and time-dependent coefficients
Introduction
The nonlinear Schrödinger equation (NLSE) with cubic and quintic nonlinearities is known to describe optical solitons in fibers and nonlinear photonic crystals [1] and matter-wave solitons in Bose-Einstein condensates [2]. The quintic nonlinearity is one among many other higher order effects that start to be important for very thin (femto-second) optical solitons and when three-body collisions in matter-wave solitons becomes non-negligable [1], [2]. The evolution of optical solitons in dispersion-managed fibers [1] and matter-wave solitons in trapped Bose-Einstein condensates [2] is described by a NLSE with a parabolic potential. Furthermore, the evolution of optical solitons in materials with complex refractive index [3], [4], [5] and matter-wave solitons in the presence of dissipation, introduces complex potentials to the NLSE. The so-called PT-symmetric potential is a particularly interesting example of such potentials. PT-symmetric potentials possess parity and time reversal symmetry. It has been shown by Bender [6] that such potentials, though complex, have real eigenvalues. A requirement for a 1D complex potential to be PT-symmetric is that its real part is an even function and its imaginary part is an odd function of the spacial variable.
It is thus experimentally-relevant to consider a NLSE with power law nonlinearities and complex potentials. Many references have studied the NLSE with cubic and quintic nonlinearities [7], [8], [9], [10], [11], [12], [13], and with parabolic [14], [15] and PT-symmetric [16], [17], [18], [19], [20], [21], [22], [23], [24] potentials using several analytical methods for nonlinear differential equations [25], [26], [27], [28], [29], [30]. We consider here a more general NLSE than what has been considered so far. The generalization is in the power of the nonlinearities not being restricted to cubic and quintic, and in introducing position- and time-dependent strengths of the dispersion and nonlinearities, and in the presence of a general complex potential (See Eq. 1 below.). A standard similarity transformation is used to transform this equation to its homogeneous version without a potential and with constant strengths of dispersion and nonlinearities. Then we use traditional methods to solve the homogeneous NLSE where we obtain a flat-top and tanh solitons. Applying the similarity transformation on these two solutions, we obtain the analogue solutions for the inhomogeneous NLSE. Further progress is made here by introducing a slightly modified similarity transformation where the solution of the homogeneous NLSE, denoted as the seed solution, is also used to construct the transformation operator, similarly to the Darboux transformation [31]. This results in solution classes composed of infinite hierarchies of exact solutions to the inhomogeneous NLSE. One class is obtained by applying the similarity transformation on all solutions of the NLSE belonging to one class. The other class is obtained by repeated action of the similarity transformation. With this modification, it became possible to find solutions for cases that were not integrable using the standard similarity transformation. One such case is the NLSE with constant coefficients and general position-dependent potential. Using the standard similarity transformation, integrability restricts the real part of the potential to a quadratic potential and the imaginary part of the potential to a constant. In the modified transformation, the real part of the potential is arbitrary and the imaginary part of the potential is related to it ensuring that when the real part is even, the imaginary part is odd, for a wide range of potentials, which allow for integrable NLSEs with PT-symmetric potential.
The rest of the paper is organized as follows. In Section 2, we derive the operator that transforms the inhomogeneous NLSE to homogeneous one. This will also result in integrability conditions that relate the coefficients of the inhomogeneous NLSE. In Section 2.1, we focus on the special case of time-dependent coefficients where we show that, for such a case, the real part of the potential is restricted to a quadratic potential with vanishing imaginary part. In Section 2.2 , we derive two exact solitonic solutions of the homogeneous NLSE. In Section 2.3, we apply the similarity transformation on these two solutions. In Section 3, we present an alternative similarity transformation and show how it leads to a hierarchy of exact solutions in the presence of PT-symmetric potentials. We end in Section 4 with conclusions and outlook.
Section snippets
Similarity transformation
The dimensionless form of the nonlinear Schrödinger’s equation (NLSE) with space- and time-dependent coefficients in dual-power law medium and external complex potential is given bywhere q(x, t) is a complex valued wavefunction, V(x, t) and W(x, t) are arbitrary real functions representing the external potential, and f(x, t) and g1, 2(x, t) are the strengths of dispersion and dual-power nonlinearities,
An alternative transformation method
Here, we present a modification on the above similarity transformation that renders it as a method of generating an infinite hierarchy of exact solutions generated from a seed solution. However, this seed solution will be used to construct the transformation. The transformation can then be used to transform other solutions of the homogeneous NLSE into those of the inhomogeneous NLSE.
We start with a similarity transformation, as given by Eq. (3)where U0(X, T) is a
Conclusion
We have derived two similarity transformations that transform a generalized inhomogeneous NLSE, Eq. (1), to a homogeneous NLSE, Eq. (2). The transformations are given by Eqs. (24) and (60). The first transformation is a standard one where solutions of the homogeneous NLSE are mapped to solutions of the inhomogeneous NLSE. The second transformation contains the additional degree of freedom of having the solution of the homogeneous NLSE as part of the operator of the transformation. In comparison
Acknowledgments
U. Al Khawaja and H.B. acknowledge the support of King Fahd University of Petroleum and Minerals under research group project RG1503-1 and RG1503-2. We also acknowledge the support of the Saudi Center for Theoretical Physics (SCTP). U. Al Khawaja Acknowledges the support of grants UAEU-UPAR(4) 2016, UAEU-UPAR(7) 2015, UAEU-UPAR(6) 2017, and RSA 2014.
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