Symmetries and reductions of integrable nonlocal partial differential equations

In this paper, symmetry analysis is extended to nonlocal differential equations, in particular for two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Obtained symmetries are used to reduce these equations to local ordinary differential equations; for the mKdV equation, without surprise, all reduced equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.


Introduction
Symmetries have been fundamentally important for understanding solutions of differential equations, e.g. [5,6,8,16]. It also reveals the integrability of partial differential equations; for instance, the Ablowitz-Ramani-Segur conjecture states that every ordinary differential equation obtained by an exact reduction of an integrable evolution equation solvable via inverse scattering transforms is of the P-type, namely without movable critical points, c.f. [4]. In this paper, we investigate integrable nonlocal partial differential equations using symmetry analysis.
The nonlocal nonlinear Schrödinger equation (NLS) i q t (x, t) + q xx (x, t) + 2q 2 (x, t)q * (−x, t) = 0, (1.1) was derived by Ablowitz and Musslimani [2] via a reduction of the AKNS system. The nonlocal NLS equation admits a great number of good properties that the classical NLS equation possesses, e.g. PT-symmetric, admitting Lax-pair and infinitely many conservation laws, solvable using inverse scattering transforms. During the latest years, integrable nonlocal systems have received great attention with many newly-proposed models, e.g. nonlocal vector NLS equation [23], multidimensional extension of nonlocal NLS equation [7], nonlocal Sine-Gordon equation, nonlinear derivative NLS equation and many other systems [1], nonlocal mKdV equation [9], Alice-Bob physics [14], nonlocal Sasa-satsuma equation [19], to mention only a few. Solutions of these systems have also been explored by many scholars; see for example, [3,9,10,[19][20][21]25]. One contrast as Ablowitz and Musslimani noticed, e.g. [1,2], is that reductions of nonlocal equations amount to nonlocal ordinary differential equations (ODEs), e.g. nonlocal Painlevé-type equations. In this paper, we show that alternative ways allow us to avoid such inconvenience. We first classify all Lie point symmetries of the nonlocal NLS equation (1.1) and the nonlocal mKdV equation [9] u Possible symmetry reductions are then conducted for both equations. While choosing the invariant variables in a proper manner, we are able to kill all nonlocal terms in the reduced ODEs. In particular for the nonlocal mKdV equation, all reduced equations are (or are equivalent to) integrable ODEs. These results are included in Section 2 and 3. In Section 4, simple transformations are defined to connect nonlocal differential equations with differential-difference equations (DDEs).

Nonlocal NLS equation
An integrable nonlocal NLS equation was proposed by Ablowitz and Musslimani [2]: where * denotes complex conjugate and q(x, t) is a complex-valued function of real variables x and t. They showed that it possesses a Lax pair and infinitely many conservation laws, and is solvable via the inverse scattering transform. We study its continuous symmetries in this section.

Lie point symmetries
As q(x, t) is complex-valued, two alternative approaches may be used to calculate its continuous symmetries. Under the coordinate (x, t, q(x, t), q * (x, t)), we consider the following local transformations For simplicity, we will omit the arguments if they are local variables (x, t). The corresponding infinitesimal generator is To apply the linearized symmetry condition [6,8,16], we adopt the following prolongation formula: It is generalised from the prolongation formula for differential equations with real variables, c.f. [11,16], but adding the conjugate terms and their prolongations. The linearized symmetry condition gives symmetries generated by the following five infinitesimal generators They can equivalently be cast into evolutionary type respectively as follows and v(x, t) are real-valued functions, and use the symmetry prolongation formula for real-valued differential equations to calculate symmetries. Now the infinitesimal generator is and the equation becomes (2.8) The following symmetries are obtained for the system above, using the linearized symmetry condition again: They correspond to the first four generators of (2.5). The last one obtained above does not appear here since it will transform the real-valued x to a complex-valued argument since ξ = i t.

Symmetry reductions
Next, we will use the symmetries to conduct possible reductions. Preferable, we choose to utilise the symmetries (2.5) with complex variables. The simplest reduction one would expect is probably traveling-wave solutions, which is difficult here as the invariant x − at becomes −x − at at (−x, t).
Consider the most general infinitesimal generator where a, b, c, d, e are arbitrary constants. The invariant variables can be found by solving the characteristic equations and we summarize the results as follows. Note that the equation depends on q(x, t) and q * (−x, t) simultaneously, and we must select the constants properly to make the invariants meaningful.
• If d = b = 0 (and a 2 + e 2 = 0), we have When a = 0 and e = 0, the reduced equation is (2.14) Since b = 0, we must choose a = e = 0. Therefore, we may simplify the invariants to y = x 2 and q(x, t) = exp(i ct)p(y). The reduced equation is Note that if we choose y = x, we will obtain a nonlocal Painlevé-type equation as shown in [2]. If we assume p(y) is real, solution of the above equation can be expressed using the Jacobi elliptic function as where C 1 , C 2 are integration constants. The above equation can actually be written as a simpler equation by introducing y = z 2 and p(z) = p(y); the resulting equation is ln |2dt − b| p(y).

(2.19)
The reduced equation is (2.20) Introducing y = z 2 and p(z) = p(y) changes the equation to

Nonlocal mKdV equation
The nonlocal mKdV equation we consider in this paper is, c.f. [9], Assuming that the infinitesimal generator reads the linearized symmetry condition gives the following generators We follow the same approach as for the nonlocal NLS equation to search for symmetry reductions. The most general symmetry generator can be denoted by where a, b, c are arbitrary constants. The characteristic equations read (3.5) • If c = 0 corresponding to the traveling-wave case, the invariants are y = bx − at and v(y) = u(x, t). (3.6) The reduced equation is When b = 0, we obtain constant solution; when b = 0, without loss of generality, it can be In principle, it can be integrated once as it admits a symmetry generated by ∂ y but it will involve the inverse of nonlocal functions. We will show some of its special solutions with the assumption a > 0.
• If c = 0, the invariants are Now we must set a = b = 0, namely reduction related to the generator −x∂ x − 3t∂ t + u∂ u . We simplify the invariants as y = t −1/3 x and v(y) = t 1/3 u(x, t) and obtain the reduced equation It can be integrated once to the second Painlevé equation Remark 3.1. As pointed out in [2] that symmetry reduction of the nonlocal NLS equation may lead to nonlocal ordinary differential equations, that is true. However, in many cases as shown from above examples, such an inconvenience can be overcome by choosing the invariant variables/functions in a proper way.
4 Transformations between nonlocal differential equations and differentialdifference equations In [24], authors introduced variable transformations to connect nonlocal and local integrable equations. For instance, the nonlocal NLS equation becomes a local NLS equation under the transformation The nonlocal complex mKdV equation become the local (classical) complex mKdV equation under the transformation In this section, we will show the relations between nonlocal differential equations and DDEs through variable transformations. For the nonlocal NLS equation, we consider the following transformations where the variable x is imaginary making x imaginary too. Let us drop the hats (always) and the nonlocal NLS equation becomes a DDE i q t + exp(−2x) (q xx − q x ) + 2q 2 q * (x + i π, t) = 0. (4.4) Let us introduce the following transformations where the variables x and t are both imaginary. The nonlocal mKdV equation becomes exp(−t)u t + exp(−x)uu(x + i π, t + i π)u x + exp(−3x) (u xxx − 3u xx + 2u x ) = 0. (4.6) Under the transformation y = exp( y), v(y) = v( y), the reduced equation (3.7) becomes b 3 exp(−2y) (v ′′′ (y) − 3v ′′ (y) + 2v ′ (y)) + (bv(y)v(y + i π) − a) v ′ (y) = 0. (4.7) These DDEs can further be rescaled and normalized. For example, taking y = i π y and v(y) = v( y), equation (4.7) becomes Similar DDEs were investigated in [18], but the variables were real-valued therein. In the same manner, the above DDEs transformed from the nonlocal NLS equation and the nonlocal mKdV equation can also be rescaled, respectively as follows: i q t + exp(− i 2πx) − 1 π 2 q xx + i π q x + 2q 2 q * (x + 1, t) = 0. (4.9) and exp(− i πt)u t + exp(− i πx)uu(x + 1, t + 1)u x + exp(− i 3πx) − 1 π 2 u xxx + 3 i π u xx + 2u x = 0. (4.10) The above examples showed that simple transformations allow us to transfer nonlocal equations to DDEs. Apparently, similar transformations can be immediately introduced for other nonlocal differential equations/systems using the same manner.

Conclusion
In this paper, symmetry analysis for integrable nonlocal differential equations was investigated; in particular, the nonlocal NLS equation and the nonlocal mKdV equation served as illustrative examples. All Lie point symmetries of these two nonlocal PDEs were obtained and possible symmetry reductions were conducted. We chose the invariant variables in a good way that the reduced ODEs are always local, at least for the two illustrative equations. At the end, we introduced some local transformations transferring nonlocal differential equations to DDEs; it is potential, hence, to extend the existing theories for DDEs to nonlocal differential equations, for instance, symmetries, conservation laws and integrability of DDEs, c.f. [12,13,15,17,18,22].