Nonautonomous "rogons" in the inhomogeneous nonlinear Schrodinger equation with variable coefficients

Nonautonomous “rogons” in the inhomoge neous nonlinear Sc hr¨ odinger equation with v ariable co efficien ts Zheny a Y an a , b a Centr o de F ´ ısic a T e´ oric a e Computacional, Universidade de Lisb o a, C omplexo Inter discipli nar, Lisb o a 1649-003, Portugal b Key L ab or atory of Mathematics Me chanization, Institute of Systems Scienc e, AMSS, Chinese A c ademy of Scienc es, Beijing 100190, Chi na Abstract The analytical nonauton omous rogons are rep orted for th e inhomogeneous nonlinear S c hr¨ odinger equation with v ariable co efficien ts in terms of rational-like functions by u sing the similarity transformation and direct ansatz. These obtained solutions can b e used to describ e the p ossible formation mechanisms for opt ical, oceanic, and matter rogue w ave phenomenon in optical fibres, the deep o cean, and Bose-Einstein condensates, respectively . Moreo ver, the snake propagation traces and th e fascinating in teractions of t wo nonautonomous rogons are generated for the chose n differen t parameters. The obtained nonautonomous rogons may excite the p ossibilit y of relative exp eriments and potential applications for the rogue wa ve phenomenon in t h e field of nonlinear science. Key wor ds: Inhomogeneous NLS eq uation with v ariable co efficien ts; Similarity transformation; Rational-lik e solutions; Rogue w av es; Rogons P ACS: 05.45.Yv; 42.65.-k; 42.8 1.Dp; 42.65. Sf; 03.75 .Lm The nonlinear Schr¨ odinger (NLS) equation is a foundational mo del describing numerous nonlinear ph ys ical phenomenon in the field o f nonlinear s cience such as optica l solitons in optical fibres [1 , 2, 3], solitons in the mean-field theory o f Bose -Einstein condensates [4, 5, 6], and the r o gu e waves (R Ws) (also known as fr e ak waves , monster waves , kil ler waves , giant waves or ex tr eme waves ) in the nonlinear o ceano graphy [7 , 8, 9, 10]. R Ws are sing le wa ves g enerated in the o cean with amplitudes m uch hig her than the av erage wa ve cr ests around them [11, 12]. Recently , the R Ws hav e attracted more and mor e attention fro m the point views o f b oth theoretical analys is [21, 22, 23, 24, 25, 28, 29, 34] and exp eriment al realization [13, 14, 30, 31, 32, 33]. The o ceanic R Ws can be, under the nonlinear theories of o c e an wa ves, modelle d by the dimensio nle s s NLS equation [7, 8] i ∂ ψ ∂ t + 1 2 ∂ 2 ψ ∂ x 2 + | ψ | 2 ψ = 0 , (1) which describes the t w o-dimensional quasi-p erio dic deep-water trains in the lowest or der in wa ve steepness and sp ectral width. In a ddition, it has b een shown that the R Ws can b e g enerated in no nlinear optical systems and the term “ optical r ogue wa ves” was c oined b y observing optical pulse pr opagation in the genera lized NLS equaiton [13, 1 4]. So me t yp es o f exact solutions o f Eq. (1) hav e been presented to describ e the p ossible formatio n mechanisms for the R W phenomenon s uc h as the algebraic breathers (Peregrine solitons) [15], the time p erio dic Email addr ess : zyy an@mmrc.i ss.ac.cn 1 breather (Ma so litons) [16], the spac e per iodic br eathers (Akhmediev breathers ) [1 7, 18]. More re c en tly , the Akhmediev breathers w ere also fur ther studied [19, 20, 21, 22]. A p ossible mechanism for the formation of R Ws was also ex hibited b y using tw o-dimensional coupled NLS eq uations [24, 25, 26] des cribing the nonlinear ly int eracting tw o -dimensional wav es in deep water. The thr e e -dimensional mechanism o f R W formation was studied in a late stage of the mo dulational instability of a per turbed Stokes deep-water wa ve [27]. F ur thermore, the optical R Ws were also found in the NLS equation with per turbing terms (the higher- o rder NLS equation with the third- order disp ersion, self-steeping, and self-frequency shift) [23]. Recen tly , the exis tence of matter R Ws in Bos e - Einstein condensa tes was pr edicted to either load into a parab olic trap or em bed in an optical lattice [34]. Here we p oint out that based on ph ysical similarities b etw een the rog ue wa ves and solitary wa ves, firs t observed by Russell in 183 4 and fur ther known as so litons by Zabusky and Krusk a l in 1965 [45], we coin the word “Ro gon” for each such “Ro gue W ave” (or the word “F reakon” for the “F rea k W av e”) if they r eappea r virtually unaffected in size or s hape shortly a fter their interactions. Similarly , there also exist the cor respo nding new terms “o ceanic rog ons”, “optica l rogons ”, and “matter r ogons” for the o ceanic ro gue wa ves [7, 8, 9 , 10], optical rogue w aves [13, 14, 21, 22, 23, 28, 29], and matter r ogue w aves [3 4], r espectively . T o the bes t o f our knowledge, ther e was no rep ort on exa ct solutions r elated to the mo dified R Ws (rogo ns) with v a riable functions b efore. In this Letter, w e will extend the NLS e quation (1) to the inhomogeneo us NLS equation with v ariable coefficients, including group-velo cit y disp ersion β ( t ), linear p otential v ( x, t ), nonlinearity g ( t ) and the g a in/loss ter m γ ( t ), in the for m [35, 36, 37, 38] i ∂ ψ ∂ t + β ( t ) 2 ∂ 2 ψ ∂ x 2 + v ( x, t ) ψ + g ( t ) | ψ | 2 ψ = iγ ( t ) ψ , (2) which is a sso ciated with δ L / δ ψ ∗ = 0, in whic h the Lagr angian density is written as L = i ( ψ t ψ ∗ − ψ ψ ∗ t ) − β ( t ) | ψ x | 2 + g ( t ) | ψ | 4 + 2[ v ( x, t ) − i γ ( t )] | ψ | 2 , whe r e ψ ≡ ψ ( x, t ), and ψ ∗ denotes the co mplex conjugate of the ph ysical field ψ . Eq. (2) can be als o k no wn as the genera lized Gross- Pitaevskii equatio n with v ariable co efficients for β ( t ) = 1 [4, 5 , 6, 44]. Our go al is fo cused on the fir s t-order and s econd-order ra tional-like so lutions of E q . (2) to describ e the p ossible formation mec hanisms of optical R Ws by using the similarity tra nsformations [39, 40, 41, 4 2, 43, 44] and direct a nsatz [21, 22]. Mor eov er, we ana ly ze the dy namical behaviors of these solutions and in tera ctions o f tw o optical R Ws (rogons ) b y choo sing the different functions. Similarity tr ansformation and n onautonomous r o gons — T o in vestigate the analytical ratio nal-like so lutions of Eq.(2) related to the optica l nonauto no mous rog ons, we employ the env elop e field ψ ( x, t ) in the gauge form [41, 42, 4 3, 4 4] ψ ( x, t ) = [Ψ R ( x, t ) + i Ψ I ( x, t )] e iϕ ( x,t ) (3) whose intensit y can b e written as | ψ ( x, t ) | 2 = | Ψ R ( x, t ) | 2 + | Ψ I ( x, t ) | 2 , wher e Ψ R ( x, t ) , Ψ I ( x, t ) and ϕ ( x, t ) are real functions o f spac e - time x, t . The substitution of tr a nsformation (3) into Eq. (2) y ie lds the system of coupled 2 real partial differen tial eq uations with v ar iable co efficients Ψ R ,t + β ( t ) 2  Ψ I ,xx + 2 ϕ x Ψ R ,x − ϕ 2 x Ψ I + ϕ xx Ψ R  + [ v ( x, t ) − ϕ t ]Ψ I + g ( t )(Ψ 2 R + Ψ 2 I )Ψ I − γ ( t )Ψ R = 0 , (4a) − Ψ I ,t + β ( t ) 2  Ψ R ,xx − 2 ϕ x Ψ I ,x − ϕ 2 x Ψ R − ϕ xx Ψ I  + [ v ( x, t ) − ϕ t ]Ψ R + g ( t )(Ψ 2 R + Ψ 2 I )Ψ R + γ ( t )Ψ I = 0 . (4b) By in tr o ducing the new v ar iables η ( x, t ) and τ ( t ), we further utilize the fo llo w ing similarity tra nsformations for the real functions Ψ R ( x, t ), Ψ I ( x, t ) and the pha se ϕ ( x, t ) Ψ R ( x, t ) = A ( t ) + B ( t ) P ( η ( x, t ) , τ ( t )) , Ψ I ( x, t ) = C ( t ) Q ( η ( x, t ) , τ ( t )) , ϕ ( x, t ) = χ ( x, t ) + µτ ( t ) , (5) to system (4) suc h that we deduce the following similar it y reduction η xx = 0 , (6a) η t + β ( t ) χ x η x = 0 , (6b) 2 χ t + β ( t ) χ 2 x − 2 v ( x, t ) = 0 , (6c) 2 σ t + [ β ( t ) χ xx − 2 γ ( t )] σ = 0 ( σ = A, B , C ) , (6d) τ t B P τ + β ( t ) 2 η 2 x C Q ηη − µτ t C Q + g ( t ) C Q [ C 2 Q 2 + ( A + B P ) 2 ] = 0 , (6e) − τ t C Q τ + β ( t ) 2 η 2 x B P ηη − µτ t ( A + B P ) + g ( t )( A + B P )[ C 2 Q 2 + ( A + B P ) 2 ] = 0 , (6f ) where µ is a co nstan t, and η ( x, t ) , χ ( x, t ) , A ( t ) , B ( t ) , C ( t ) , τ ( t ) , P ( η, τ ) , Q ( η, τ ) are functions to b e deter- mined. After some algebr a, it follows from Eqs. (6a)-(6d) that we ha ve η ( x, t ) = α ( t ) x + δ ( t ) , χ ( x, t ) = − α t 2 α ( t ) β ( t ) x 2 − δ t α ( t ) β ( t ) x + χ 0 ( t ) , v ( x, t ) = χ t + β ( t ) 2 χ 2 x , (7a) A ( t ) = a 0 p | α ( t ) | e R t 0 γ ( s ) ds , B ( t ) = bA ( t ) , C ( t ) = cA ( t ) , (7b) where a 0 , b, c a re consta n ts, α ( t ) (the inv erse of the wa ve width), δ ( t ) ( − δ ( t ) /α ( t ) b eing the p osition of its center of mass), and χ 0 ( t ) are all free functions o f time t . T o further reduce Eqs. (6e) a nd (6 f) to the system o f coupled pa r tial differential eq uations with co nstan t co efficien ts, W e require the conditions: τ t = β ( t ) 2 η 2 x and g ( t ) = β ( t ) 2 GA − 2 ( t ) η 2 x ( G = const . ), which can g enerate the constraints for the v ar iable τ ( t ) a nd nonlinear it y g ( t ) τ ( t ) = 1 2 Z t 0 α 2 ( s ) β ( s ) ds, g ( t ) = Gα ( t ) β ( t ) 2 a 2 0 e 2 R t 0 γ ( s ) ds (8) such that Eqs. (6e) a nd (6f) reduce to the coupled system of differential equation with constant co efficien ts bP τ + cQ ηη − µcQ + cGQ [ c 2 Q 2 + (1 + bP ) 2 ] = 0 , (9a) − cQ τ + bP ηη − µ (1 + bP ) + G (1 + bP )[ c 2 Q 2 + (1 + b P ) 2 ] = 0 . (9b) 3 F ollowing the direct approach develop ed in Refs. [28, 29], w e can o bta in the rational solutions of s ystem (9) such that the corres p onding rational-like solutions (nonautonomous rogons) of Eq. (2) can be found in terms of similarity tra ns formations (3) and (5). In the following, we will exhibit the dy namical behaviors of rational-like solutions with man y in teres ting nontrivial featur e s. First-or der r ational-like solution– It follows from s ystem (9) that w e hav e the solutio n P ( η , τ ) = − 4 / [ b A 1 ( η , τ )] and Q ( η , τ ) = − 8 τ / [ c A 1 ( η , τ )] with A 1 ( η , τ ) = 1 + 2 η 2 + 4 τ 2 for µ = G = 1. Th us ba s ed on the similar it y transformatio ns (3) and (5), w e obtain the first-order rational- lik e s olution (nona uto nomous r ogon) of Eq. (2) ψ 1 ( x, t ) = a 0 p | α ( t ) | e R t 0 γ ( s ) ds  1 − 4 + 8 iτ ( t ) 1 + 2[ α ( t ) x + δ ( t )] 2 + 4 τ 2 ( t )  e i [ χ ( x,t )+ τ ( t )] , (10) whose in tensit y c a n be written as | ψ 1 ( x, t ) | 2 = a 2 0 | α ( t ) | e 2 R t 0 γ ( s ) ds  2[ α ( t ) x + δ ( t )] 2 + 4 τ 2 ( t ) − 3  2 + 6 4 τ 2 ( t ) { 1 + 2[ α ( t ) x + δ ( t )] 2 + 4 τ 2 ( t ) } 2 , (11) where χ ( x, t ) and τ ( t ) are given b y E qs. (7a) and (8 ). It is easy to s ee that the rational-like solution (10) is differen t from the known rational solution of the NLS equation (1) [15, 16, 21, 22, 2 3, 28, 29], since it contains so me free functions of time t , which will g enerate abundant structur es rela ted to the optical R Ws. In par ticular, when α = 2 , a 0 = β = 1 , χ 0 = γ = 0 , resulting in g = 1, Eq. (2 ) r educes to E q. (1) such that the no nautonomous rogo n solution (10) reduce s to the known rogon so lution in Refs. [2 1, 28]. In what follows, we will choos e some free functions of time to ex hibit the obtained rational-like solution (10). F or the fixed para meters α 0 = 1 , χ 0 ( t ) = 0 and γ ( t ) = 0 . 1 tanh( t )sech( t ) , i) if w e c ho ose o ther free functions as the p olynomials of time t , i.e. α ( t ) = 1 , β ( t ) = 0 . 5 t 2 , then Figures 1 a and 1b depict the dynamical behavior of the rationa l-like so lution (10) for differen t terms δ ( t ) = t, t 2 , resp ectively , in which the other co efficients g ( t ) and v ( x, t ) in E q. (2) are given by g ( t ) = 1 4 t 2 e [sech ( t ) − 1] / 5 , (12a) v ( x, t ) = 4 t − 3 x + t − 2 for δ ( t ) = t, (12b) v ( x, t ) = 4 t − 2 x + 4 for δ ( t ) = t 2 ; (12c) ii) if we cho o se other free functions as the p erio dic functions of time t , i.e. α ( t ) = dn( t, k ) , β ( t ) = cn( t, k ), then Figures 2a and 2b display the dynamical b ehavior of the in tensity of the r ational-like so lutio n (10) for different terms δ ( t ) = sn( t, k ) , cn( t, k ), resp ectively , in which the co efficients g ( t ) and v ( x, t ) in E q. (2) are giv en by g ( t ) = 1 2 cn( t, k ) dn( t, k ) e [sech( t ) − 1] / 5 , (13a) v ( x, t ) = k 2 cn( t, k ) 2 dn 2 ( t, k ) x 2 + 1 2 cn( t, k )[ k 2 sd( t, k ) x − 1] 2 for δ ( t ) = sn( t, k ) , (13b) v ( x, t ) = k 2 cn( t, k ) 2 dn 2 ( t, k ) x 2 + dn( t, k ) cn 2 ( t, k ) x + 1 2 cn( t, k )[ k 2 sd( t, k ) x + sc( t, k )] 2 for δ ( t ) = cn( t, k ); (13c) 4 iii) if we choose other free functions as the p erio dic functions of time t , i.e. α ( t ) = cn( t, k ) , β ( t ) = dn( t, k ), then Figur es 3a a nd 3b exhibit the dynamical b ehavior of the rational-like so lution (10) for different ter ms δ ( t ) = sn( t, k ) , dn( t, k ), res pectively , in which the co efficients g ( t ) and v ( x, t ) in Eq. (2) ar e given by g ( t ) = 1 2 cn( t, k ) dn( t, k ) e [sech( t ) − 1] / 5 , (14a) v ( x, t ) = dn( t, k ) 2 cn 2 ( t, k ) x 2 + 1 2 dn( t, k )[sc( t, k ) x − 1] 2 for δ ( t ) = sn( t, k ) , (14b) v ( x, t ) = k 2 dn( t, k ) 2 cn 2 ( t, k ) x 2 + k 2 cn( t, k ) dn 2 ( t, k ) x + 1 2 dn( t, k )[sc( t, k ) x + k 2 sd( t, k )] 2 for δ ( t ) = dn( t, k ) . (14c) It follows from these figures that the rational-like solution (10) is different from the known r ational solution o f the NLS equation (1) [15, 16, 21, 22, 2 3, 28, 29], and may b e useful to raise the p ossibility of relative exp eriment s and potential applications for the R W phenomenon. Se c ond-or der r ational-like s olution — It follows from (9) that w e hav e the second-or der rationa l s olution of Eq. (2) in the form P ( η , τ ) = P 2 ( η , τ ) / [ b A 2 ( η , τ )] a nd Q ( η , τ ) = Q 2 ( η , τ ) / [ c A 2 ( η , τ )], w he r e P 2 ( η ( x, t ) , τ ( t )) = − 1 2 η 4 − 6 η 2 τ 2 − 1 0 τ 4 − 3 2 η 2 − 9 τ 2 + 3 8 , Q 2 ( η ( x, t ) , τ ( t )) = − τ  η 4 + 4 η 2 τ 2 + 4 τ 4 − 3 η 2 + 2 τ 2 − 15 4  , A 2 ( η ( x, t ) , τ ( t )) = 1 12 η 6 + 1 2 η 4 τ 2 + η 2 τ 4 + 2 3 τ 6 + 1 8 η 4 + 9 2 τ 4 − 3 2 η 2 τ 2 + 9 16 η 2 + 33 8 τ 2 + 3 32 , (15) As a consequence, based on the transfor mations (3) and (5), we have the s econd-order rationa l-lik e so lution (t wo-rog on solution) of Eq. (2) ψ 2 ( x, t ) = a 0 p | α ( t ) | e R t 0 γ ( s ) ds  1 + P 2 ( η ( x, t ) , τ ( t )) + i Q 2 ( η ( x, t ) , τ ( t )) A 2 ( η ( x, t ) , τ ( t ))  e i [ χ ( x,t )+ τ ( t )] (16) whose in tensit y is g iv en by | ψ 2 ( x, t ) | 2 = a 2 0 | α ( t ) | e 2 R t 0 γ ( s ) ds [ A 2 ( η ( x, t ) , τ ( t )) + P 2 ( η ( x, t ) , τ ( t ))] 2 + Q 2 2 ( η ( x, t ) , τ ( t )) A 2 2 ( η ( x, t ) , τ ( t )) , (17) where η ( x, t ) = α ( t ) x + δ ( t ), and χ ( x, t ) , τ ( t ) ar e g iv en b y Eqs. (7a) and (8). The o btained second-or der rational-like solution (16) is also differ en t from the known rational solutions of the NLS eq uation (1) [21, 28], since some free functions o f time t a re inv o lv e d. Similar to the first-o rder rationa l-like s olution, the solution (16) can a lso r educe to the known one of Eq. (1) in Refs. [21, 28]. F or the fixed parameters α 0 = 1 , χ 0 ( t ) = 0 , γ ( t ) = 0 . 1 tanh( t )sec h( t ), i) if we cho ose other free functions as the po lynomials of time t , i.e . α ( t ) = 1 , β ( t ) = 0 . 5 t 2 , then Figures 4a and 4 b depict the dyna mical interaction of the tw o -rogon solution (16) for differen t terms δ ( t ) = 0 . 1 t, t 2 ; ii) if we c ho ose the free functions as the p erio dic functions of time t , i.e. α ( t ) = dn( t, k ) , β ( t ) = cn( t, k ), then Figures 5a and 5b illustra te the dynamical int eraction of the tw o- rogon solution (16) fo r different terms δ ( t ) = sn( t, k ) , cn( t, k ); iii) if we ta ke other free functions as the p erio dic functions of time t , i.e. α ( t ) = cn( t, k ) , β ( t ) = dn( t, k ), then Figures 6a a nd 6b exhibit the dynamical in ter action o f the tw o-r ogon solution (16) for differ en t terms δ ( t ) = sn( t, k ) , dn( t, k ). 5 In c o nclusion, we hav e presented the a nalytical first-or der and s econd-order rationa l-like solution pairs (nonautonomous rogons) of the inhomogeneo us nonlinear Schr¨ o dinger equation with v a r iable co efficients b y using the similarity transfo r mation and direct ansa tz. By using the dire c t a pproach [28], we can also obtain the higher-or der rational-like solutions of Eq. (2) which are omitted her e. These o btained so lutions b e used to describ e the p ossible formation mechanisms for the o ptical ro gue w av e phenomenon in optical fibres , the o ceanic rog ue wa ve phenomenon in the deep o cean, and the matter rogue w av e phenomenon in Bose-E instein condensates. F urthermore, it should b e empha s ized that we give the explicit so lutions for the existence of matter rogo ns in Bose- Einstein condensates . Moreov er, the sna k e pr opagation trac e s and the interaction of optical nonautono mous ro gons a re exhibited by cho osing some free functions of time. Some sha pes of the optical nonautonomous r ogons a nd fascina ting interactions b et ween tw o optical nonautonomous r ogons are also achiev ed with different functions. These solutions mo dify the known solutions rela ted to the optical r o gons. 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Konotop, Ph ys. Rev. E 80 (2009) 036607 . [45] N. J. Zabusky , M. D. Kruska l, Phys. Rev . Lett . 15 (1965) 240. 7 List of the F igure Captions Figure 1. W av e pro pagations (left column ) and co n tour plots (right column) for the intensit y | ψ 1 | 2 (11) of the first- order rational-like solution (10 for α = α 0 = 1 . 0 , β = 0 . 5 t 2 , γ ( t ) = 0 . 1 tanh( t )sec h( t ). (a)-(b) δ ( t ) = t ; (c)-(d) δ ( t ) = t 2 . Figure 2. W av e propagations (left c o lumn) and contour plots (right column) for the intensit y | ψ 1 | 2 (11) of the first- order rationa l-lik e solution (10) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α = dn( t, k ) , β ( t ) = cn( t, k ): (a)-(b) δ ( t ) = sn( t, k ); (c)-(d) δ ( t ) = cn( t, k ). Figure 3. W av e propagations (left c o lumn) and contour plots (right column) for the intensit y | ψ 1 | 2 (11) of the first- order rationa l-lik e so lutio n (10) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sec h( t ) , k = 0 . 6 , α = cn( t, k ) , β ( t ) = dn( t, k ): (a)-(b) δ ( t ) = sn( t, k ); (c)-(d) δ ( t ) = dn( t, k ). Figure 4. W av e propag ations (left column) and contour plots (right column) for the intensit y | ψ 2 | 2 (17)of the seco nd- order rationa l-lik e solution (16) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α = dn( t, k ) , β ( t ) = cn( t, k ): (a)-(b) δ ( t ) = 0 . 1 t ; (c)-(d) δ ( t ) = t 2 . Figure 5. W av e pro pagations (left c o lumn) a nd contour plots (right co lumn) for the intensit y | ψ 2 | 2 (17) of the second- order rationa l-lik e solution (16) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α = dn( t, k ) , β ( t ) = cn( t, k ): (a)-(b) δ ( t ) = sn( t, k ); (c)-(d) δ ( t ) = cn( t, k ). Figure 6. W av e pro pagations (left c o lumn) a nd contour plots (right co lumn) for the intensit y | ψ 2 | 2 (17) of the second- order rationa l-lik e so lutio n (16) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sec h( t ) , k = 0 . 6 , α = cn( t, k ) , β ( t ) = dn( t, k ): (a)-(b) δ ( t ) = sn( t, k ); (c)-(d) δ ( t ) = dn( t, k ). 8 Figure 1: ( color online). W a ve propagations (left column) and con tour plots (right column) for the intensit y | ψ 1 | 2 (11) of the first-order rational-like solution ( 10) for α = α 0 = 1 . 0 , β = 0 . 5 t 2 , γ ( t ) = 0 . 1 tan h ( t )sec h( t ). (a)-( b ) δ ( t ) = t ; (c)-(d) δ ( t ) = t 2 . Figure 2: (color online). W av e propagations (left column) and contour plots (right column) for the intensity | ψ 1 | 2 (11) of the fi rst-order rational-like solution (10) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α ( t ) = dn( t, k ) , β ( t ) = cn( t, k ): (a)-(b) δ ( t ) = sn( t , k ); (c)-(d) δ ( t ) = cn( t , k ) . 9 Figure 3: (color online). W av e propagations (left column) and contour plots (right column) for the intensity | ψ 1 | 2 (11) of the fi rst-order rational-like solution (10) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α ( t ) = cn( t, k ) , β ( t ) = dn( t, k ): (a)-(b) δ ( t ) = sn( t , k ); (c)-(d) δ ( t ) = dn ( t, k ). Figure 4: (color online). W av e propagations (left column) and contour plots (right column) for the intensity | ψ 2 | 2 (17) of the second-order rational-lik e solution (16) for α = α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , β ( t ) = 0 . 5 t 2 : (a)-(b) δ ( t ) = 0 . 1 t ; (c)-(d) δ ( t ) = t 2 . 10 Figure 5: ( color online). W a ve propagations (left column) and con tour plots (right column) for the intensit y | ψ 2 | 2 (17) of the second-order rational-lik e solution (16 ) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α ( t ) = dn( t , k ) , β ( t ) = cn( t, k ): (a)-( b ) δ ( t ) = sn ( t, k ); (c)-(d) δ ( t ) = cn( t, k ). Figure 6: ( color online). W a ve propagations (left column) and con tour plots (right column) for the intensit y | ψ 2 | 2 (17) of the second-order rational-like solution (16) for α 0 = 1 . 0 , γ ( t ) = 0 . 1 tanh( t )sech( t ) , k = 0 . 6 , α ( t ) = cn( t, k ) , β ( t ) = dn( t, k ): ( a)- (b) δ ( t ) = sn( t, k ); (c)- ( d) δ ( t ) = dn( t, k ). 11