Symmetry broken and symmetry preserving multi-soliton solutions for nonlocal complex short pulse equation

Chaos, Solitons and Fractals 130 (2020) 109451

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Symmetry broken and symmetry preserving multi-soliton solutions for nonlocal complex short pulse equation H. Sarfraz∗, U. Saleem Department of Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan

a r t i c l e

i n f o

Article history: Received 28 June 2019 Revised 27 August 2019 Accepted 15 September 2019

Keywords: Reverse space-time nonlocal complex short pulse equation (NL-CSPE) Symmetry preserving and symmetry non-preserving solutions Wadati–Konno–Ichikawa (WKI) scheme Darboux transformation (DT) Quasideterminants

a b s t r a c t In this article we consider a general system of complex short pulse equation (CSPE) that under certain nonlocal symmetry reduction yields a reverse space-time nonlocal complex short pulse equation (NLCSPE). We apply matrix Darboux transformation to the associated Lax pair and construct multi-soliton solutions. K-soliton solution is expressed in terms of quasideterminant formula which enable us to compute explicit expressions of symmetry broken and symmetry preserving one- and two-soliton solutions for NL-CSPE. In addition to these solutions, we also obtain one-bright and interaction of two-bright solitons for classical CSPE. All these investigations end up with the conclusion that both symmetry preserving and broken solutions exist for NL-CSPE. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Schäfer and Wayne introduced an integrable equation called short pulse equation (SPE)

qxt = q +

1  3 q , xx 6

(1.1)

where the real-valued function q = q(x, t ) denotes the magnitude of electric field. SPE (1.1) describes the propagation of optical short pulses of the order of femto seconds (ultra pulses) in nonlinear media and can also be associated with pseudospherical surfaces [1–4]. The integrability of SPE (1.1) has been investigated through various view points, such as, existence of Lax pair of Wadati–Konno–Ichikawa (WKI) type [5], existence of bi-Hamiltonian structure [6], conservation laws [7,8], soliton solutions etc [8–14]. In recent years PT -symmetric integrable systems have attained great consideration [15–21]. They are physically significant because of their applications in various fields, e.g, lattice dynamics, nonlinear optics, acoustics etc [22–28]. To understand the origin of PT -symmetry, we have to revisit an important concept in quantum mechanics. In quantum mechanics only Hermitian Hamiltonian can give rise to real eigenvalues. But Bender and Boettcher [29] have conjectured that in general, a non-Hermitian Hamilto-

nian can also lead to a real eigenvalue spectrum provided it respects the PT -symmetry. Violation of above symmetry is called spontaneous PT -symmetry breaking, which occurs when imaginary part of potential exceeds a certain threshold value. Above the critical value, eigenvalue spectrum ceases to be real. Therefore we may say that these critical points are onset of phase transition from PT -exact phase (or symmetry preserving phase) to PT broken phase. Such symmetry breaking takes place in the vicinity of exceptional points where eigenvalue branches coalesce and phase transition takes place. Existence of exceptional points has been confirmed experimentally [30]. In 2013, Ablowitz and Musslimani have presented a nonlocal nonlinear Schrödinger equation and solved it via inverse scattering transform method [15]. They have shown that nonlocal NLS equation is completely integrable. In [31], the authors proposed some other nonlocal integrable equations, e.g, nonlocal sineGordon equation, nonlocal 3-wave equation, nonlocal modified Korteweg-de Vries (mKdV) equation, nonlocal Davey Stewartson (DS) equation. Recently, a nonlocal reverse space-time short-pulse equation has been presented in [32,33]. In order to study NL-CSPE we would like to initiate with a general system of coupled nonlinear evolution equations, i.e.

qxt + q −

1 (qrqx )x = 0, 2

(1.2)

rxt + r −

1 (rqrx )x = 0, 2

(1.3)

∗

Corresponding author. E-mail addresses: [email protected] (H. Sarfraz), [email protected] (U. Saleem). https://doi.org/10.1016/j.chaos.2019.109451 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

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H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

where q(x, t) and r(x, t) are complex-valued scalar functions. System (1.2) and (1.3) is in fact the compatibility condition of the following Wadati–Konno–Ichikawa (WKI) scheme [5]:

x ≡ M = (−iλσ3 + λQx ),

(1.4)

  i t ≡ N = − σ3 + N0 + λN1 , 4λ where



Q=

 N1 =





0 q

r , 0

−

iqr 2

qrqx 2

N0 = rqrx 2

iqr

ir

0 − iq

 ,

σ3 =

2



2

,

0

2



(1.5)

1

0

0

−1

(1.6)

 .

(1.7)

Under local symmetry reduction which is, r = −q∗ , the coupled Eqs. (1.2) and (1.3) yield classical complex short pulse equation (CSPE)

qxt + q +

1 2  |q| qx x = 0. 2

(1.8)

Furthermore, if we take q∗ = q, we obtain SPE. Similarly under nonlocal symmetry reduction, r (x, t ) = −q∗ (−x, −t ), the system (1.2) and (1.3) reduces to NL-CSPE given by [32,33], i.e.,

qxt (x, t ) + q(x, t ) +

1 (q(x, t )q∗ (−x, −t )qx (x, t ) )x = 0, 2

(1.9)

1 q∗xt (−x, −t ) + q∗ (−x, −t ) + (q∗ (−x, −t )q(x, t )q∗x (−x, −t ) )x = 0. 2 (1.10) It can be clearly seen that the Eqs. (1.9) and (1.10) are invariant under the joint action of t → −t, x → −x and complex conjugation. Also the potential term V (x, t ) = −q(x, t )q∗ (−x, −t ) is obeying the PT -symmetry, i.e., V (x, t ) = V ∗ (−x, −t ). From (1.9) and(1.10) the standard classical CSPE (1.8) can be recovered by simply replacing q∗ (−x, −t ) by q∗ (x, t). To our knowledge, the dynamics of multisoliton solution of NL-CSPE (1.9) and (1.10) has never been studied in recent years. As far as nonlocal nonlinear Schrödinger equation is concerned different types of solutions have been computed by using different techniques. Therefore, it is natural to study the soliton dynamics of various solutions of NL-CSPE (1.9) and (1.10). In this paper we would like to compute symmetry broken and symmetry preserving solutions for NL-CSPE (1.9) and (1.10). Nonlocal integrable systems are more interesting than their local counterparts as they may possess both symmetry non-preserving and symmetry preserving solutions at the same time. In general, a system obeying PT -symmetry may contain both symmetry broken and symmetry preserving solutions simultaneously. Simplest examples of such models are the well-known nonlocal nonlinear Schrödinger equation, nonlocal Manakov system etc [17,18]. In NL-CSPE (1.9) and (1.10) the dynamical variables q(x, t) and q∗ (−x, −t ) evolve independently i.e. if q(x, t) is evaluated at (x, t) then q∗ (−x, −t ) has to be evaluated at (−x, −t ). If we choose entirely distinct eigenvalues then the solutions q(x, t) and r (x, t ) ≡ −q∗ (−x, −t ) could no longer be deduced from one another and therefore such solutions are regarded as symmetry non-preserving (or broken) solutions of NL-CSPE (1.9) and (1.10). For such case the corresponding potential function V(x, t) will be complex-valued. However when eigenvalues are chosen with certain coherence, then the dynamical variables will no longer remain independent and can be deduced from each other under joint action of parity, time-reversal and complex conjugation operations. Such solutions are called symmetry preserving solutions and possess real-valued

potential. As soon as eigenvalues are taken as conjugate pair the symmetry preserving solutions of NL-CSPE (1.9) and (1.10) would reduce to that of classical CSPE (1.8). The plan of this paper is as follows. In Section 2, we shall present a quick review of Darboux transformation and compute multi-soliton solutions in terms of quasideterminants of particular solutions to the associated WKI scheme. In Section 3, we shall compute explicit expressions of symmetry non-preserving and symmetry preserving one- and two-soliton solutions for NLCSPE (1.9) and (1.10). We shall analyze different dynamics of symmetry non-preserving and symmetry preserving soliton solutions of NL-CSPE (1.9) and (1.10). Moreover, one and two-soliton solutions of classical CSPE (1.8) would be obtained from that of symmetry preserving solutions of NL-CSPE (1.9) and (1.10), under suitable reductions on spectral parameters. At the end concluding remakes would be given in Section 4. 2. Darboux transformation Darboux transformation is an important algebraic method to construct new solutions of a second order differential equation from a given solution. The particular applications of the Darboux transformation is the Sturm–Liouville equation [11]. In the recent decades, DT has also been studied for a large class of nonlinear integrable PDE’s (partial differential equations) which admit a linear problem. DT on the matrix-valued solution  of the WKI scheme (1.4) and (1.5) is defined as

  [1] = λ−1 I − P ,

(2.1)

where I = diag(1,1) and P = H −1 H −1 . Here H is the particular matrix solution of the linear system (1.4) and (1.5) and  is an invertible matrix of eigenvalues. The action of DT (2.1) generate one-fold transformed matrix-valued function as

N0[1] = N0 −

i σ3 , H −1 H −1 . 4

(2.2)

Using the properties of quasideterminants1 Eq. (2.2) may be expressed as

N0[1] = N0 +

i 4



  H σ3 ,  (1) H

    = N0 + i σ3 , (1) ,  O2 4 I2

where H (1 ) = H −1 . Also



(1 )

   =  (1) 

  (1 )   = 11 O2   (1 ) I2

21

(1 ) 12 (1 ) 22

(2.5)

 ,

(2.6)

along with  = H and (1 ) = H (1 ) . From Eq. (2.5) one-fold dressed dynamical variables become (1 ) q[1] (x, t ) = q(x, t ) − 21 ,

(1 ) r [1] (x, t ) = r (x, t ) − 12 .

(2.7)

1 Quasideterminants are counterparts of ordinary determinants but with noncommuting entries defined over non-commutative ring. To understand quasideterminants we take D = [d f g ] a square matrix having non-commuting entries. Then by definition of quasideterminant |D|mn is written as

|D|mn = dmn − rmn (Dmn )−1 cnm ,

(2.3)

n and cnm denotes mth row and nth column vectors respectively of D-matrix where rm obtained by removing dmn entry. Also a submatrix Dmn is obtainable by removing mth row and nth column of D-matrix respectively. Quasideterminant |D|mn can also be expressed as ratio of ordinary determinants as

|D|mn = (−1 )m+n

det D . det Dmn

For more details see reference [34,35].

(2.4)

H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

The action of K-times DT on matrix-valued function N0 becomes

  H1  ⎢  H1(1) ⎢  . i⎢  N0[K] = N0 + ⎢σ3 ,  .. 4⎢  ⎣ H1(K−1)  (K )  H1 ⎡

··· ··· .. .

H2 H2(1) .. . H2(K−1)

HK HK(1) .. . HK(K−1)

···

(K )

(K )

···

H2

HK

⎤   ⎥ ⎥ ⎥ ⎥, ⎥ ⎦  O2  O2 O2 .. . I2

(2.8) ( j)

where Hm =

N0[K] = N0 +

−j Hm m ,

we can re-write Eq. (2.8) as

i σ3 ,  ( K ) , 4

where

   (K ) =  (K ) 

  (K )  = 11 O2   (K )

(K ) 12



(K )

22

21

,

(2.10)

H3

···

HK

H2(1)

H3(1)

···

HK(1)

H2(2)

H3(2)

···

HK(2)

.. .

.. .

..

.. .

H2(K−1)

H3(K−1)

···

⎜ H (1 ) ⎜ 1 ⎜ (2 ) ⎜  = ⎜ H1 ⎜ . ⎜ . ⎝ . H1(K−1)



(K ) = H1(K )



F ( K ) = O2 and

⎞

H2

H1

H2(K ) O2

O2

H3(K ) ···

I2

T

.

(3.3)

3.1. Single soliton solutions From (2.7), we have (1 ) q[1] (x, t ) = −21 ,

(1 ) r [1] (x, t ) = −12 .

(3.4)



 ≡ H1 = 

(1 )

(1 )

≡ H1

X1

X2

Y1

Y2





,

−1 1 =

 −1 λ1 X1 = λ−1 Y1 1

 λ−1 X2 2 , λ−1 Y2 2

λ−1 1

0



λ−1 2

0

,

F (1) = I2 .

HK

⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

  X1 F2(1)    =  Y1 O2   −1 λ1 X1

    (1 ) 12 =  (1) 1 (2.11)

,

X2 Y2

λ−1 X2 2

(3.6)

    λ−1 X 2 2  , X2  Y2  X2

(3.7)

(2.12)

,

  X1   0  λ−1 X  1 =− 1 1  X1  0  Y1

(3.5)

  X1 F1(1)    =  Y1 O2   −1 λ1 Y1

    (1 ) 21 =  (1) 2

(2.13)

 −1 λ Y1   1     = − Y1  X1  0  Y1

X2 Y2

1 0

λ−1 Y2 2

 λ−1 Y2  2 Y2   . X2  Y2  (3.8)





X2m−1 Y2m−1

Hm =

X2m , Y2m

m =

 λ2m−1 0

0

λ2m

 ,

( j) j Hm = Hm − m .

Upon substituting explicit expressions of Xi and Yi into (3.7) and (3.8) and substituting the resulting expressions in Eq. (3.4), we get

(2.14) Eq. (2.9) allows us to write K-fold transformed scalar dynamical variables as (K ) q[K] (x, t ) = q(x, t ) − 21 ,

(2.15)

(K ) r [K] (x, t ) = r (x, t ) − 12 .

(2.16)

The above expressions play an important role in the construction of symmetry broken and symmetry preserving solutions of NLCSPE (1.9) and (1.10) and as well as stable soliton solutions of classical CSPE (1.8). In what follows, we would like to calculate explicit expressions of one- and two-soliton solutions in vanishing background. 3. Symmetry non-preserving and preserving soliton solutions In order to calculate explicit expressions of single and double soliton solutions, we take zero as seed solution i.e. q(x, t ) = r (x, t ) = 0. In zero background WKI scheme (1.4) and (1.5) becomes



−iλ

x =

0

 t =

Y (x, t ) = Be−θ (x,t ) ,

where θ (x, t ) = −iλx − 4iλ t, here A and B are the constants (real or complex-valued).

HK(K−1)

 (K )

···

X (x, t ) = Aeθ (x,t ) ,

(1 ) (1 ) The matrix element 12 and 21 are given by

along with

⎛

Y )T . Integration of above linear system yields

For single soliton solution, we have

(2.9)

F (K ) 

where  = (X

3

− 4iλ 0

 , iλ  0

0 i

4λ

,

(3.1)

(3.2)

(λ1 − λ2 ) AB22 e−2θ2 (λ1 − λ2 ) AB22 e−θ1 −θ2 − 2   , =− e2(θ1 −θ2 )+R − 1 λ1 λ2 2λ1 λ2 sinh θ1 − θ2 + R2 R

q[1] (x, t ) = − 

(3.9)

−q[1]∗ (−x, −t ) = − 

(λ1 − λ2 ) AB11 e2θ1  1 − e2(θ1 −θ2 )+R λ1 λ2

(λ1 − λ2 ) AB11 eθ1 +θ2 − 2   ≡ r[1] (x, t ), 2λ1 λ2 sinh θ1 − θ2 + R2 R

=

(3.10)

A B

where R = ln( A1 B2 ). Eqs. (3.9) and (3.10) allow us to compute more 2 1

general solutions of general CSPE (1.2) and (1.3). We obtain tallest singular solutions if we take distinct values of all parameters. In order to obtain symmetry non-preserving solutions for NL-CSPE (1.9) and (1.10), we take A1 = −A2 = B1 = B2 = 1 in Eqs. (3.9) and (3.10) with distinct spectral parameters. PT symmetry is said to be broken because the obtained solutions (3.9) and (3.10) are not parity-time reversal transformed complex conjugate of each other, i.e., solution (3.9) cannot be recovered from (3.10) under the combined action of parity, time-reversal and complex conjugation, also the product of dynamical variables, i.e., potential term V [1] (x, t ) = −q[1] (x, t )q[1]∗ (−x, −t ) is complexvalued. Different profiles of symmetry non-preserving solutions (3.9) and (3.10) are shown in Fig. 1 for the parameters λ1 = 0.9 + i and λ2 = −0.5 − i. Fig. 1 shows oscillating singular type solutions

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H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

Fig. 1. Profiles of symmetry non-preserving one-soliton solutions (3.9) and (3.10) of NL-CSPE (1.9) and (1.10).

with loss and gain along positive t-directions as shown in Fig. 1a and 1b respectively. It is mandatory to again mention here that potential V(x, t) is complex-valued for above choice and hence solutions are termed as symmetry broken solutions. We would like to compare our results at this stage with that of nonlocal nonlinear Schrödinger (NNLS) equation. We obtain singular type structure with gain and loss as symmetry broken solutions for NL-CSPE which are contrary to NNLS equation where unstable smooth solitons were obtained as symmetry broken solutions [17]. In order to obtain symmetry preserving solutions of NL-CSPE (1.9) and (1.10) we consider A1 = −A2 = B1 = B2 = 1 and λ1 = iδ1 and λ2 = −iδ 1 . Here δ 1 and δ 1 are positive real constants and δ1 = δ 1 (here over-bar does not stand for complex conjugation). For this choice, Eqs. (3.9) and (3.10) become,



q[1] (x, t ) = −



i



δ1 + δ 1 e

δ1 δ 1 1 + e (

2x δ 1 −

)

2x δ1 +δ 1 − 2t





−q[1]∗ (−x, −t ) = −

δ1 δ 1

i

t 2δ 1





1 δ1 + δ 1

δ1 + δ 1 e

( 1+e

1

3.2. Double-soliton solutions In zero background two-soliton solutions become (2 ) q[2] (x, t ) = −21 ,

(2 ) r [2] (x, t ) = −12 .

(3.14)

In this case, we have

 ≡

 ,

(3.11)

H1 (1 )

H1



H2 (1 )

H2

⎛

X1

⎜ Y1 ⎝λ−1 X1 1 λ−1 Y1 1

=⎜

X2

X3

X4

Y2 −1 X2 2 −1 Y2 2

Y3 −1 X3 3 −1 Y3 3

Y4 −1 X4 4 −1 Y4 4

λ λ

λ λ

λ λ

⎞ ⎟ ⎟, ⎠ (3.15)

2xδ1 − 2δt

1

)

2x δ1 +δ 1 − 2t



1 1 δ1 + δ 1

 .

(3.12)

This time spectral parameters are so smartly chosen that they tend the dynamical variables to preserve the nonlocal symmetry that means now both the solutions (3.11) and (3.12) can be deduced from one another under parity, time reversal and complex conjugation. This time symmetry is also said to be preserved because the potential V(x, t) becomes real-valued. Profiles of symmetry preserving soliton solutions (3.11) and (3.12) of NL-CSPE (1.9) and (1.10) are shown in Fig. 2 for parameters δ1 = 0.5, δ 1 = 0.7, which show unstable solitons with gain and loss along positive t-direction as shown in Figs. 2a and 2b respectively. Fig. 2c has been plotted for |V(x, t)| that indicates stable soliton due to balanced gain and loss. At this stage NNLS equation possesses oscillatory type solutions [17]. It is worthwhile to mention that stable one-soliton solution is obtained at λ2 = λ∗1 (called exceptional point). At this stage eigenvalues are related and corresponding eigenvectors become parallel to each other. In other words this stable soliton solution of classical CSPE (1.8) can be obtained from symmetry preserving solutions of NL-CSPE (1.9) and (1.10), given by Eqs. (3.11) and (3.12), if we take δ1 = δ 1 , i.e.,

 t  i q[1] (x, t ) = − sech − 2xδ1 . δ1 2δ1

Such points are called EPs at which NL-CSPE (1.9) and (1.10) reduces to that of classical CSPE (1.8). Profile of one-bright soliton solution (3.13) for classical CSPE (1.8) is shown in Fig. 3 for δ1 = 1.

(3.13)

 (2) ≡ H1(2)



H2(2) =

 −2 λ1 X1 λ−2 Y1 1

λ−2 X2 2 λ−2 Y2 2

λ−2 X3 3 λ−2 Y3 3

 λ−2 X4 4 , λ−2 Y4 4 (3.16)



F ( 2 ) = O2

I2

T

.

(3.17)

(2 ) (2 ) The matrix entries 12 and 21 are

  X1   Y1  −1 (2 ) 12 = λ1 X1  λ−1  1 Y1 λ−2 X 1 1

X2 Y2

X3 Y3

X4 Y4

λ−1 X2 2 λ−1 Y2 2 −2 λ2 X2

λ−1 X3 3 λ−1 Y3 3 −2 λ3 X3

λ−1 X4 4 λ−1 Y4 4 −2 λ4 X4

X2 Y2

X3 Y3

X4 Y4

λ−1 X2 2 λ−1 Y2 2 −2 λ2 Y2

λ−1 X3 3 λ−1 Y3 3 −2 λ3 Y3

λ−1 X4 4 λ−1 Y4 4 −2 λ4 Y4

and

  X1   Y1  −1 (2 ) 21 = λ1 X1  λ−1  1 Y1  λ−2Y 1 1

     0 ,  1   0  0 0

     1 .  0   0 

(3.18)

0 0

(3.19)

H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

5

Fig. 2. Profiles of symmetry preserving one-soliton solutions (3.11) and (3.12) of NL-CSPE (1.9) and (1.10).

Fig. 3. Profile of one-bright soliton solution (3.13) of classical CSPE (1.8).

and

Eq. (3.14) becomes

  X1   Y1  −2 λ Y1  1 λ−1Y1 [2] q (x, t ) =  1  X1   Y1  −1 λ X1  1  λ−1Y1 1

X2 Y2

X3 Y3

λ−2 Y2 2 λ−1 Y2 2

λ−2 Y3 3 λ−1 Y3 3

X2 Y2

X3 Y3

λ−1 X2 2 λ−1 Y2 2

λ−1 X3 3 λ−1 Y3 3

     −2  λ4 Y4  λ−1 Y4  4 , X4   Y4   λ−1 X4  4 λ−1 Y4  4 X4 Y4

(3.20)

  X1   Y1  −1 λ X1  1 λ−2 X1 [2] r (x, t ) =  1  X1   Y1  −1 λ X1  1  λ−1Y1 1

X2 Y2

X3 Y3

λ−1 X2 2 λ−2 X2 2

λ−1 X3 3 λ−2 X3 3

X2 Y2

X3 Y3

λ−1 X2 2 λ−1 Y2 2

λ−1 X3 3 λ−1 Y3 3

     −1 λ4 X4  λ−2 X4  4 . X4   Y4   λ−1 X4  4 λ−1 Y4  4 X4 Y4

(3.21)

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H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

Fig. 4. Interaction of symmetry non-preserving two-soliton solution (3.20) and (3.21) of NL-CSPE (1.9) and (1.10).

Fig. 5. Profiles of symmetry preserving two-soliton solutions (3.20) and (3.21) of NL-CSPE (1.9) and (1.10).

For A1 = −A2 = B1 = B2 = 1, A3 = −A4 = B3 = B4 = 1 by keeping the spectral parameters entirely distinct in Eqs. (3.20) and (3.21) we have symmetry non-preserving two-soliton solutions of NL-CSPE (1.9) and (1.10). The profiles of symmetry broken twosoliton solutions (3.20) and (3.21) for NL-CSPE (1.9) and (1.10) are shown in Fig. 4 for spectral parameters λ1 = 0.1 + i, λ2 = −0.5 − i, λ3 = 0.3 + 0.5i and λ4 = −0.8 − 0.5i. Fig. 4 shows interaction of two-singular structures with gain and loss. For symmetry preserving solutions of NL-CSPE (1.9) and (1.10), we take A1 = −A2 = B1 = B2 = 1, A3 = −A4 = B3 = B4 = 1, λ1 = iδ1 ,

λ2 = −iδ 1 , λ3 = iδ2 , λ4 = −iδ 2 , (where δi = δ i ) in Eqs. (3.20) and (3.21). The profiles of symmetry preserving two-soliton solutions (3.20) and (3.21) for NL-CSPE (1.9) and (1.10) are shown in Fig. 5 for δ1 = 0.2, δ 1 = 0.28, δ2 = 0.5 and δ 2 = 0.58 which show interactions of unstable growing and decaying solitons and interaction of two stable solitons (for potential term) as shown in Figs. 5a–5c respectively. To construct two-soliton solution of classical CSPE (1.8), we take δi = δ i , i.e., λ2k = λ∗2k−1 . Profile of twosoliton solution (3.20) for classical CSPE (1.8) is shown in Fig. 6 for parameters δ1 = 1, δ2 = 0.5.

H. Sarfraz and U. Saleem / Chaos, Solitons and Fractals 130 (2020) 109451

7

Fig. 6. Profiles of two-soliton solution (3.20) of classical CSPE (1.8).

4. Conclusion In this work, we have started with a general complex short pulse equation that turns into a reverse space-time nonlocal complex short pulse equation (NL-CSPE) upon applying suitable nonlocal symmetry reduction. We have also studied the Darboux transformation and obtained a general expression of K-soliton solutions in terms of quasideterminants. Finally, we have used quasideterminant formula to obtain exact expressions of one- and two-soliton solutions. We have computed symmetry non-preserving and symmetry preserving soliton solutions for particular values of spectral parameters. When eigenvalues are differently chosen then dynamical variables cannot be related and hence leading to symmetry non-preserving solutions for NL-CSPE. For such solutions potential term is complex-valued. However when eigenvalues are related with a certain coherence, the resulting dynamical variables become symmetric and those solutions are called symmetry preserving solutions of NL-CSPE. For symmetry preserving solutions potential is real-valued. Dynamics of one- and two symmetry non-preserving solutions (growing and decaying oscillatory solutions) and symmetry preserving solutions (unstable soliton with gain/loss) are plotted for defined choice of spectral parameters. Furthermore bright one-soliton and interaction of two bright-solitons for classical CSPE are also incorporated in this work. It would be interesting to study the multi-component NL-CSPE and their soliton solutions in future work. Furthermore, these investigations can be further extended to study rogue wave solutions of NL-CSPE. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement We would like to express our deepest gratitude for the reviewers on their valuable comments. H. S. gratefully acknowledges Higher Education Commission (HEC) Pakistan for provision of support via Indigenous Scholarship for Ph.D. studies, Phase 2, Batch 3, 2015. References [1] Rothenberg JE. Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses. Opt Lett 1992;17:1340.

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