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With the aid of a direct projective approach, a general transformation solution for the nonautonomous nonlinear Schr?dinger (NLS) system is derived. Based on certain known exact solutions of the projective equation, some periodic and localized excitations with novel properties are correspondingly revealed by entrancing appropriate system parameters. The integrable constraint conditions for the nonautonomous NLS system derived naturally here are consistent with the compatibility condition via the Painlevé analysis in other literature.

Nonlinear science is believed, by many outstanding scientists, to be the most deeply important frontier for understanding nature and applications in reality. For example, nonlinear optical solitons are regarded as the natural data bits and an important alternative for the next generation of ultrahigh speed optical telecommunication systems. It is known that the propagation of electromagnetic waves in nonlinear optical waveguides and the ground state wave functions of Bose-Einstein condenstates (BECs) can be described by nonlinear Schrödinger (NLS) system [

where is the complex wave envelope in a comoving frame, with time-modulated dispersion, nonlinearity, net gain (>0) or loss (<0), and external harmonic trap potential in BECs, which is usually absent in nonlinear optical transmission aspect [

In recent decades, many powerful approaches have been devised, such as inverse scattering theory, Bäcklund transformation, Hirota’s bilinear method, Darboux transformation, the hyperbola function method, the mixed exponential method, homogeneous balance method, and multilinear variable separation approach et al. [

where,

,

are polynomials of and their derivatives, (T indicates the transposition of a matrix). By using a travelling wave transformation based on the following ansatz, where are real arbitrary constants, and substituting the ansatz into Equation (2) yields an ordinary differential equation system. Then is expanded into a polynomial in

where are constants to be determined and N is fixed by balancing the linear term of the highest order with the nonlinear term in Equation (2). If we suppose , and or respectively, then the corresponding approaches are usually called the tanh-function method, the sech-function method and the Jacobian-function method. Although the Jacobian elliptic function method is more improved than the tanh-function method and the sech-function method, the repeated calculations are often tedious since the different function should be treated in a repeated way. The idea of deformation projective approach is that, is not assumed to be a specific function, such as tanh, sech, sn and cn, etc., but a solution of the projective equation, such as the Riccati equation, (is a constant and the prime denotes differentiation with respect to), the cubic nonlinear KG equation, (here c, and are all arbitrary constants), or the Jacobian elliptic equation. Using the projective relation (3) and the solutions of the related projective equations, one can obtain many explicit and exact travelling wave solutions of system (2). Along with this line, some scholars further assume that the projective function is an exact excitation of the generalized Jacobian elliptic equation where are all arbitrary constants. Because the generalized Jacobian elliptic equation possesses more exact solutions [

Recently, the above projective transformation approach is further extended for finding novel localized excitations of a physical model [15-17]. With the help of projective transformation idea and based on the general reduction theory, the projective algorithm is extended that: For the system (2) we assume its solution in an extended symmetric form

where, are arbitrary functions to be determined, is a solution of the Riccati equation or the generalized Jacobian elliptic equation. N is also determined by balancing the highest nonlinear terms and the highest-order partial terms in the given nonlinear system. Substituting the ansatz (4) together with the related projective equations into Equation (2), collecting coefficients of polynomials of, then setting each coefficient to zero, yields a set of partial differential equations concerning and. Solving the system of partial differential equations to obtain and, substituting the derived results and the solutions of the related projective equations into Equation (4), one can derive many exact solutions to the given nonlinear system.

Motivated by the above ideas, one may assume a selfsimilar family model as the projective equation, which has been extensively studied in other literature [18,19]. For instant, when discussing a nonautonomous Korteweg-de Vries (KdV) system, we may use the classical (1 + 1)- dimensional KdV equation as a projective equation since the autonomous KdV equation has been widely explored. In the following part of the paper, the (1 + 1)-dimensional nonautonomous NLS system is selected to illustrate the selfsimilarity projective approach, and a general exact solution to the nonautonomous NLS system is derived, which can convert all exact solutions of self-similar well-known standard nonlinear Schröding equation to the corresponding exact solutions of the nonautonomous NLS system.

The standard nonlinear Schrödinger (NLS) equation can be used as a self-similar projective equation of the nonautonomous NLS system assumed, the dimensionless autonomous NLS form is

where are system parameters, is a complex wave function of related system dynamics in physical fields such as BBCs and nonlinear optics. As is known, the nature of NLS equation has been widely explored via different approaches before, and its exact solutions are vast in related literature. For example, when and, the NLS equation possesses a canonical bright soliton solution, while and, the NLS equation has a fundamental dark soliton solution

.

For more detailed information, one may refer to a review in reference [

In order to build a direct projective relation between the nonautonomous NLS system (1) and the standard NLS Equation (5), we construct an ansatz to the nonautonomous NLS system as follows

where is an exact complex solution to the standard NLS Equation (5), are similarity variables, is a similarity wave phase, and is a time-dependent function to modulate the wave amplitude to be determined. The reason for to be factored out of in format (6) is the existence of the coefficients functions of time t in Equation (1). All the parameters all real differential functions, and should be well chosen to avoid some singularity of the complex wave function. Substituting Equation (6) together with Equation (5) into the nonautonomous NLS system (1), collecting coefficients of polynomials of and its derivatives, then setting each coefficient to zero, yields a set of partial differential equations

After some careful and direct algebra, one can obtains the amplitude, self-similarity variables and phase of the complex wave pulse

where are integrable constants, is defined by

and with a constraint condition

It is interesting to note that when the net gain coefficient, the constraint condition (18) becomes

which is just the completely integrable compatibility condition via the Painlevé analysis [4,7], i.e., a subtle balance condition to keep the nonautonomous NLS system integrable.

From the management viewpoint of the solitons, Equation (18) provides an effectively way to manipulate soliton dynamics. When any three parameters among and are set, the remaining one can be tuned correspondingly via Eqaution (18) to control the coherent dynamics of solitons. Actually, some special applications of equation (18) have been explored in reference [

where

and are arbitrary constants, dn(·, m) denotes the dn type Jacobian elliptic function with being its modulus. This leads to the following family of double periodic wave solutions to the nonautonomous NLS system (1)

where

and the integrable constants in Equations (13)-(16) are set to be zero for simplicity. In the limit case, the above double periodic wave solution (22) will be reduced to a solitary wave solution

with the constraint condition (18). As a special case, if and, the constrain condition (18) becomes

where is a integrable constant, then the above corresponding exact solutions read

where

with

Once the nonlinear parameter is explicitly given, then all the exact solutions and their related dynamic behaviors of the nonautonomous NLS system are correspondingly determined. For integrality and readability, some remarks are given as follows:

Firstly, as a special situation, if, and, i.e., in Equation (26), the nonautonomous NLS system (1) has the canonical soliton solutions regardless of the explicit form of the timedependent nonlinearity and dispersion. This is because in this case the the constraint condition is identified and the scale factor is a constant. In this sense the soliton solution of Equation (1) is a fundamental canonical soliton. While, i.e., in Equation (26), the original balance between nonlinearity and dispersion is broken down. In the case the canonical soliton must deform itself to build new balance between nonlinearity and dispersion, and the soliton-like solution of Equation (1) is a deformed canonical soliton or a similariton due to the amplitude of the soliton scaled by the factor. And it actually means that if and, the exact solitons of the nonautonomous NLS system (1) are quasi-canonical or deformed solitons depending on if is equal to or not as mentioned above.

Secondly, for more general cases, if and, the constraint condition (18) indicates that, which means the amplitude of the soliton must vary with factor. This leads to an important and a novel phenomenon that there does not exist the canonical and even quasi-canonical matter-wave solitons under the constraint condition (18). However, when we set, (is an arbitrary constant), i.e., ,

and is tuned by the constraint condition (18) as follows

then we can rederive fundamental canonical solitons, which distinctly indicates the influence of the dispersion, nonlinear managements and net gain to the localized excitation behaviors.

Finally, it is also interesting to mention that the external trap potential is absent in the selfsimilarity projective transformations (13)-(16). However, the presence of the potential affects the balance between nonlinearity, dispersion and net gain via the constraint condition (18), and builds a deep relation between the optical solitons and the matter-waves.

In summary, the direct self-similarity projective approach is successfully applied to the nonautonomous nonlinear Schrödinger system. In terms of the known exact solutions of the self-similarity projective equation, i.e., the standard nonlinear Schröding equation, some significant types of localized excitations with novel properties are correspondingly revealed by entrancing appropriate system parameters. The present analysis can be applied to all exact solutions of the nonautonomous nonlinear Schrödinger system. The self-similarity projective approach provides an effective and a systematical way to investigate the nonlinear dynamics of the nonautonomous nonlinear Schrödinger system. By the way, as a comparison it is helpful to mention some techniques to find the localized excitation solutions of the nonautonomous NLS equation in previous literature. The Lax pair analysis is very useful in discussing integrability conditions. And a widely used approach is the deformation projective method, which introduces some explicit transformation parameters. These parameters are determined by a set of partial differential equations, which in general case are little solved analytically as emphasized in reference [

The authors are in debt to Professors Y. M. Liu and J. F. Zhang, Doctors C. Q. Dai and T. T. Jia for their fruitful discussions. The work was supported by the National Natural Science Foundation of China under Grant No.11172181, the Natural Science Foundation of Guangdong Province of China under Grant No. 101512005- 01000008, the Special Foundation of Talent Engineering of Guangdong Province of China, and the Scientific Research Foundation of Key Discipline of Guangdong Shaoguan University.