Riemann-Hilbert approach and N-soliton solution for an eighth-order nonlinear Schrodinger equation in an optical fiber
aa r X i v : . [ m a t h - ph ] O c t Riemann-Hilbert approach and N -soliton solution for aneighth-order nonlinear Schr¨odinger equation in an opticalfiber Zhou-Zheng Kang , , Tie-Cheng Xia ∗
1. Department of Mathematics, Shanghai University, Shanghai 200444, China;2. College of Mathematics, Inner Mongolia University for Nationalities,Tongliao 028043, China
Abstract
This paper aims to present an application of Riemann-Hilbert approach to treat higher-order nonlinear differential equation that is an eighth-order nonlinear Schr¨odinger equationarising in an optical fiber. Starting from the spectral analysis of the Lax pair, a Riemann-Hilbert problem is formulated. Then by solving the obtained Riemann-Hilbert problemunder the reflectionless case, N -soliton solution is generated for the eighth-order nonlinearSchr¨odinger equation. Finally, the three-dimensional plots and two-dimensional curves withspecific choices of the involved parameters are made to show the localized structures anddynamic behaviors of one- and two-soliton solutions. AMS Subject classification : 35C08
Keywords : eighth-order nonlinear Schr¨odinger equation; Riemann-Hilbert approach; solitonsolutions
In this paper, we investigate in detail an eighth-order nonlinear Schr¨odinger (NLS) equation iq t + A K [ q ( x, t )] − iA K [ q ( x, t )] + A K [ q ( x, t )] − iA K [ q ( x, t )]+ A K [ q ( x, t )] − iA K [ q ( x, t )] + A K [ q ( x, t )] = 0 , (1)which is used for describing the propagation of ultrashort nonlinear pulses. , It can be generatedfrom truncating the infinite hierarchy of nonlinear Schr¨odinger equations that is used to investigatethe higher-order dispersive effects and nonlinearity. Here q ( x, t ) denotes a normalized complex ∗ Corresponding author. E-mail: [email protected]. q ( x, t ) mean the partial derivativeswith respect to the scaled spatial coordinate x and time coordinate t correspondingly. Eachcoefficient A j (2 ≤ j ≤
8) is an arbitrary real number, and K [ q ( x, t )] = q xx + 2 q | q | ,K [ q ( x, t )] = q xxx + 6 | q | q x ,K [ q ( x, t )] = q xxxx + 6 q ∗ q x + 4 q | q x | + 8 | q | q xx + 2 q q ∗ xx + 6 | q | q,K [ q ( x, t )] = q xxxxx + 10 | q | q xxx + 30 | q | q x + 10 qq x q ∗ xx + 10 qq ∗ x q xx + 20 q ∗ q x q xx + 10 q x q ∗ x ,K [ q ( x, t )] = q xxxxxx + q (cid:2) | q x | q ∗ + 50 q xx ( q ∗ ) + 2 q ∗ xxxx (cid:3) + q (cid:2) q xxxx q ∗ + 8 q x q ∗ xxx + 22 | q xx | + 18 q xxx q ∗ x + 70 q x ( q ∗ ) (cid:3) + 20 q x q ∗ xx + 10 q x (cid:0) q xx q ∗ x + 3 q xxx q ∗ (cid:1) + 20 q xx q ∗ + 10 q (cid:2) ( q ∗ x ) + 2 q ∗ q ∗ xx (cid:3) + 20 q | q | ,K [ q ( x, t )] = q xxxxxxx + 70 q xx q ∗ x + 112 q x | q xx | + 98 q xxx | q x | + 70 q (cid:2) q x ( q ∗ x ) + 2 q x q ∗ q ∗ xx + q ∗ (cid:0) q xx q ∗ x + q xxx q ∗ (cid:1)(cid:3) + 28 q x q ∗ xxx + 14 q (cid:2) q ∗ (cid:0) | q x | q x + q xxxxx (cid:1) + 3 q xxx q ∗ xx + 2 q xx q ∗ xxx + 2 q x q ∗ xxxx + 20 q x q xx ( q ∗ ) (cid:3) + 140 | q | q x + 70 q x ( q ∗ ) + 14 q ∗ (5 q xx q xxx + 3 q x q xxxx ) .K [ q ( x, t )] = q xxxxxxxx + 14 q (cid:2) | q x | ( q ∗ ) + 20 q xx ( q ∗ ) + 2 q ∗ xxxx q ∗ + 3( q ∗ xx ) + 4 q ∗ x q ∗ xxx (cid:3) + q (cid:2) q ∗ (14 q xx q ∗ xx + 11 q xxx q ∗ x + 6 q x q ∗ xxx ) + 238 q xx ( q ∗ x ) + 336 | q x | q ∗ xx + 560 q x ( q ∗ ) + 98 q xxxx ( q ∗ ) + 2 q ∗ xxxxxx (cid:3) + 2 q (cid:8) q x (cid:2) q ∗ x ) + 14 q ∗ q ∗ xx (cid:3) + q x (cid:2) q xx q ∗ x q ∗ + 238 q xxx ( q ∗ ) + 6 q ∗ xxxxx (cid:3) + 34 | q xxx | + 36 q xxxx q ∗ xx + 22 q xx q ∗ xxxx + 20 q xxxxx q ∗ x + 161 q xx ( q ∗ ) + 8 q xxxxxx q ∗ (cid:9) + 182 q xx | q xx | + 308 q xx q xxx q ∗ x + 252 q x q xxx q ∗ xx + 196 q x q xx q ∗ xxx + 168 q x q xxxx q ∗ x + 42 q x q ∗ xxxx + 14 q ∗ (cid:0) q x q ∗ x + 4 q xxxxx q x + 5 q xxx + 8 q xx q xxxx (cid:1) + 490 q x q xx ( q ∗ ) + 140 q q ∗ (cid:2) ( q ∗ x ) + q ∗ q ∗ xx (cid:3) + 7 q | q | . Here the superscript ∗ represents complex conjugate.As a matter of fact, Equation (1) covers many nonlinear differential equations of importantsignificance, some of which are listed as follows:(i) For the case of A = A = A = A = A = A = 0, Equation (1) is reduced to thefundamental nonlinear Schr¨odinger equation describing the propagation of the picosecond pulsesin an optical fiber.(ii) For the case of A = and A = A = A = A = A = 0, Equation (1) is reduced tothe Hirota equation , describing the third-order dispersion and time-delay correction to the cubicnonlinearity in ocean waves.(iii) For the case of A = and A = A = A = A = 0, Equation (1) becomes a fourth-orderdispersive NLS equation , describing the ultrashort optical-pulse propagation in a long-distance,high-speed optical fiber transmission system. 2iv) For the case of A = and A = A = A = 0, Equation (1) becomes a fifth-order NLSequation describing the attosecond pulses in an optical fiber.By now, there have been plenty of researches on Equation (1). For instance, the interactionsamong multiple solitons were under study, and oscillations in the interaction zones were observedsystematically. As a result, it was found that the oscillations in the solitonic interaction zonespossess different forms with different spectral parameters of Equation (1) and so forth. In afollow-up study, the Lax pair and infinitely-many conservation laws were derived via symboliccomputation, which verifies the integrability of Equation (1). Moreover, the one-, two- and three-soliton solutions were explored as well by means of the Darboux transformation.The principal aim of this study is to determine multi-soliton solutions for the eighth-order NLSequation (1) with the aid of the Riemann-Hilbert approach. − This paper is divided into fivesections. In second section, we recall the Lax pair associated with Equation (1) and convert it intoa desired form. In third section, we carry out the spectral analysis, from which a Riemann-Hilbertproblem is set up on the real line. In fourth section, the construction of multi-soliton solutions forEquation (1) is detailedly discussed in the framework of the Riemann-Hilbert problem under thereflectionless case. A brief conclusion is given in the final section.
Upon the Ablowitz-Kaup-Newell-Segur formalism, the eighth-order NLS equation (1) admits a2 × Ψ x = U Ψ , U = i ς q ∗ q − ς , (2a)Ψ t = V Ψ , V = X j =0 ς j a j b j c j − a j , (2b)where Ψ = (Ψ , Ψ ) T is a vector eigenfunction, Ψ and Ψ are the complex functions of x and t ,the symbol T signifies transpose of the vector, and ς is a isospectral parameter. Furthermore,3 = − iA (cid:8) | q | + 21 q xx ( q ∗ ) − | q x | + 14 q ∗ q ∗ xx + 70 q q ∗ (( q ∗ x ) + q ∗ q ∗ xx ) − q xxx q ∗ xxx + q ∗ xx q xxxx + 7 q (cid:2) q ∗ q ∗ xxxx + 3( q ∗ xx ) + 4 q ∗ x q ∗ xxx + 10 q x ( q ∗ ) q ∗ x + 10 q xx ( q ∗ ) (cid:3) − q xxxxx q ∗ x + q x (28 q ∗ q ∗ x q xx + 28( q ∗ ) q xxx − q ∗ xxxxx )+ q ∗ q xxxxxx + q (cid:2) q ∗ ) ( q x ) + 14( q ∗ x ) q xx + 28 q x q ∗ x q ∗ xx + 14 q ∗ x (4 q xx q ∗ xx + q ∗ x q xxx + q x q ∗ xxx ) + 14( q ∗ ) q xxxx + q ∗ xxxxxx (cid:9) + A (cid:8) − q ( q ∗ ) q ∗ x + 20( q ∗ ) q x q xx + q ∗ xx q xxx + 10 q (cid:2) q ∗ ) q x − q ∗ x q ∗ xx − q ∗ q ∗ xxx (cid:3) − q xx q ∗ xxx − q ∗ x q xxxx + q x q ∗ xxxx + q ∗ (10 q x q ∗ x + q xxxxx ) − q (cid:2) q ∗ q ∗ x q xx + 10 q x (( q ∗ x ) − q ∗ q ∗ xx ) (cid:3) − q ∗ ) q xxx + q ∗ xxxxx (cid:9) − iA (cid:8) | q | + 5( q ∗ ) q xx + q xx q ∗ xx + 5 q (cid:2) ( q ∗ x ) + 2 q ∗ q ∗ xx (cid:3) − q ∗ x q xxx − q x q ∗ xxx + q ∗ x q xxxx + q (cid:2) q ∗ ) q xx + q ∗ xxxx (cid:3)(cid:9) + A (cid:8) − q q ∗ q ∗ x − q ∗ x q xx + q x q ∗ xx + q ∗ q xxx + q (cid:2) q ∗ ) q x − q ∗ xxx (cid:3)(cid:9) − iA (cid:8) q ( q ∗ ) − q x q ∗ x + q ∗ q xx + qq ∗ xx (cid:9) + A ( q ∗ q x − qq ∗ x ) − iA qq ∗ ,a = 2 A (cid:8) q xxx ( q ∗ ) q ∗ x − q ∗ ) q x q xx − q ∗ xx q xxx + 10 q ( − q ∗ ) q x + 2 q ∗ x q ∗ xx + q ∗ q ∗ xxx ) + q xx q ∗ xxx q ∗ x q xxxx − q x q ∗ xxxx − q ∗ (10 q x q ∗ x + q xxxxx ) + q (cid:2) q ∗ q ∗ x q xx + 10 q x (( q ∗ x ) − q ∗ q ∗ xx ) − q ∗ ) q xxx + q ∗ xxxxx (cid:3)(cid:9) − iA (cid:8) | q | + 5( q ∗ ) q x + q xx q ∗ xx + 5 q (( q ∗ x ) + 2 q ∗ q ∗ xx ) − q ∗ x q xxx − q x q ∗ xxx + q ∗ q xxxx + q (10( q ∗ ) q xx + q ∗ xxxx ) (cid:9) + 2 A (cid:8) q q ∗ q ∗ x + q ∗ x q xx − q x q ∗ xx − q ∗ x q xxx + q ( − q ∗ x ) q x + q ∗ xxx ) (cid:9) − iA (cid:8) | q | − q x q ∗ x + q ∗ q xx + qq ∗ xx (cid:9) + 2 A (cid:8) q x q ∗ + qq ∗ x (cid:9) − iA qq ∗ ,a = 4 iA (cid:8) | q | + 5( q ∗ ) q x + q xx q ∗ xx + 5 q (( q ∗ x ) + 2 q ∗ q ∗ xx ) − q ∗ x q xxx − q x q ∗ xxx + q ∗ q xxxx + q (10( q ∗ ) q xx + q ∗ xxxx ) (cid:9) + 4 A (cid:8) q q ∗ q ∗ x + q ∗ x q xx − q x q ∗ xx − q ∗ q xxx + q ( − q ∗ ) q x + q ∗ xxx ) (cid:9) + 4 iA (cid:8) | q | − q x q ∗ x + q ∗ q xx + qq ∗ xx (cid:9) + 4 A (cid:8) q x q ∗ − qq ∗ x (cid:9) + 4 iA qq ∗ + 2 iA ,a = − A (cid:8) q q ∗ q ∗ x + q ∗ x q xx − q x q ∗ xx − q ∗ q xxx + q ( − q ∗ ) q x + q ∗ xxx ) (cid:9) + 8 iA (cid:8) q ( q ∗ ) − q x q ∗ x + q xx q ∗ + qq ∗ xx (cid:9) − A (cid:8) − q x q ∗ + qq ∗ x (cid:9) + 8 iA qq ∗ + 4 iA ,a = − iA (cid:8) | q | − q x q ∗ x + q ∗ q xx + qq ∗ xx (cid:9) + 16 A (cid:8) q x q ∗ − qq ∗ x (cid:9) − iA qq ∗ − iA ,a = 32 A (cid:8) − q x q ∗ + qq ∗ x (cid:9) − iA qq ∗ − iA ,a = 64 iA qq ∗ + 32 iA , a = 64 iA , a = − iA , = A (cid:8) | q | q x + 70( q ∗ ) q x + 70 q xx q ∗ x + 112 q x q xx q ∗ xx + 98 q x q xxx q ∗ x + 70 q ( q x ( q ∗ x ) + 2 q ∗ q ∗ xx ) + q ∗ (2 q xx q ∗ x + q ∗ q xxx ) + 28 q x q ∗ xxx + 14 q ∗ (5 q xx q xxx + 3 q x q xxxx ) + 14 q (20( q ∗ ) q x q xx + 3 q ∗ xx q xxx + q xx q ∗ xxx + 2 q ∗ x q xxxx + q x q ∗ xxxx + q ∗ (20 q x q ∗ x + q xxxxx )) + q xxxxxxx (cid:9) + iA (cid:8) q ( q ∗ ) + 20 q ∗ q xx + 20 q x q ∗ xx + 10 q (5 q ∗ x q xx + 3 q xxx q ∗ ) + 2 q (35( q ∗ ) q x + 11 q xx q ∗ xx + 9 q ∗ q xxx + 4 q x q ∗ xxx + 6 q xxxx q ∗ ) + 2 q (30 q ∗ q x q ∗ x + 25( q ∗ ) q xx + q ∗ xxxx )+ q xxxxxx (cid:9) + A (cid:8) | q | q x + 10 q x q ∗ x + 20 q ∗ q x q xx + 10 q ( q ∗ x q xx + q x q ∗ xx + q ∗ q xxx ) + q xxxxx (cid:9) + iA (cid:8) q | q | + 6 q ∗ q x + 4 q ( q x q ∗ x + 2 q ∗ q xx ) + 2 q q ∗ xx + q xxxx (cid:9) + A (cid:8) qq ∗ q x + q xxx (cid:9) + iA (cid:8) q q ∗ + q xx (cid:9) + A q x ,b = − iA (cid:8) q | q | + 20 q ∗ q xx + 20 q x q ∗ xx + 10 q (( q ∗ x ) + 2 q ∗ q ∗ xx ) + 10 q x (5 q ∗ x q xx + 3 q ∗ x q xxx ) + 2 q (35( q ∗ q x ) + 11 q xx q ∗ xx + 9 q ∗ x q xxx + 4 q x q ∗ xxx + 6 q ∗ q xxxx )+ 2 q (30 q ∗ q x q ∗ x + 25( q ∗ ) q xx + q ∗ xxxx ) + q xxxxxx (cid:9) + 2 A (cid:8) | q | q x + 10 q x q ∗ x + 20 q ∗ q x q xx + 10 q ( q ∗ x q xx + q x q ∗ xx + q ∗ q xxx ) + q xxxxx (cid:9) − iA (cid:8) q | q | + 6 q ∗ q x + 4 q ( q x q ∗ x + 2 q ∗ q xx ) + 2 q q ∗ xx + q xxxx (cid:9) + 2 A (cid:8) qq ∗ q x + q xxx (cid:9) − iA (cid:8) q q ∗ + q xx (cid:9) + 2 A q x − iA q,b = − A (cid:8) | q | q x + 10 q x q ∗ x + 20 q ∗ q x q xx + 10 q ( q ∗ x q xx + q x q ∗ xx + q ∗ q xxx )+ q xxxxx (cid:9) − iA (cid:8) q | q | + 6 q ∗ q x + 4 q ( q x q ∗ x + 2 q ∗ q xx ) + 2 q q ∗ xx + q xxxx (cid:9) − A (cid:8) | q | q x + q xxx (cid:9) − iA (cid:8) q | q | + q xx (cid:9) − A q x − iA q,b = 8 iA (cid:8) q | q | + 6 q ∗ q x + 4 q ( q x q ∗ x + 2 q ∗ q xx ) + 2 q q ∗ xx + q xxxx (cid:9) − A (cid:8) | q | q x + q xxx (cid:9) + 8 iA (cid:8) q | q | + q xx (cid:9) − A q x + 8 iA q,b = 16 A (cid:8) | q | q x + q xxx (cid:9) + 16 iA (cid:8) q | q | + q xx (cid:9) + 16 A q x + 16 iA q,b = − iA (cid:8) q | q | + q xx (cid:9) + 32 A q x − iA q, b = − A q x − iA q,b = 128 iA q, b = 0 , c j = b ∗ j . Let us now rewrite the Lax pair (2) in a more convenient formΨ x = i ( ςσ + Q )Ψ , (3a)Ψ t = (cid:2) i (cid:0) A ς + 4 A ς − A ς − A ς + 32 A ς + 64 A ς − A ς (cid:1) σ + Q (cid:3) Ψ , (3b)where σ = − , Q = q ∗ q ,Q = (cid:0) a + a ς + ˆ a ς + ˆ a ς + ˆ a ς + ˆ a ς + ˆ a ς (cid:1) σ + X j =0 ς j b j c j , a l mean q ( x, t ) and its derivative terms appeared in a l (2 ≤ l ≤ In this section, we focus on putting forward a Riemann-Hilbert problem for the eighth-order NLSequation (1). Now we assume that the potential function q ( x, t ) in the Lax pair (3) decays to zerosufficiently fast as x → ±∞ . It can be known from (3) that when x → ±∞ ,Ψ ∝ e iςσx + i (2 A ς +4 A ς − A ς − A ς +32 A ς +64 A ς − A ζ ) σt , which motivates us to introduce the variable transformationΨ = µ e iςσx + i (2 A ς +4 A ς − A ς − A ς +32 A ς +64 A ς − A ς ) σt . Upon this transformation, the Lax pair (3) can be changed into the desired form µ x = iς [ σ, µ ] + U µ, (4a) µ t = i (cid:0) A ς + 4 A ς − A ς − A ς + 32 A ς + 64 A ς − A ς (cid:1) [ σ, µ ] + Q µ, (4b)where [ · , · ] is the matrix commutator and U = iQ. From (4), we find that tr( U ) = tr( Q ) = 0.In the direct scattering process, we will concentrate on the spectral problem (4a), and the t -dependence will be suppressed. We first introduce two matrix Jost solutions µ ± of (4a) expressedas a collection of columns µ − = ([ µ − ] , [ µ − ] ) , µ + = ([ µ + ] , [ µ + ] ) (5)meeting the asymptotic conditions µ − → I , x → −∞ , (6a) µ + → I , x → + ∞ . (6b)Here the subscripts of µ indicated refer to which end of the x -axis the boundary conditions arerequired for, and I stands for the identity matrix of size 2. Actually, the solutions µ ± are uniquelydetermined by the integral equations of Volterra type µ − = I + Z x −∞ e iςσ ( x − y ) U ( y ) µ − ( y, ς )e iςσ ( y − x ) d y, (7a) µ + = I − Z + ∞ x e iςσ ( x − y ) U ( y ) µ + ( y, ς )e iςσ ( y − x ) d y. (7b)After direct analysis on Equations (7) we can see that [ µ − ] , [ µ + ] are analytic for ς ∈ C − and6ontinuous for ς ∈ C − ∪ R , while [ µ + ] , [ µ − ] are analytic for ς ∈ C + and continuous for ς ∈ C + ∪ R ,where C − and C + are respectively the lower and upper half ς -plane: C − = { ς ∈ C | Im( ς ) < } , C + = { ς ∈ C | Im( ς ) > } . Next we set out to study the properties of µ ± . In fact, it can be shown from tr( U ) = 0 thatthe determinants of µ ± are independent of the variable x . Evaluating det µ − at x = −∞ anddet µ + at x = + ∞ , we get det µ ± = 1 for ς ∈ R . In addition, µ − E and µ + E are both fundamentalsolutions of (3a), where E = e iςσx , they are linearly dependent µ − E = µ + ES ( ς ) , ς ∈ R . (8)Here S ( ς ) = ( s kj ) × is called the scattering matrix and det S ( ς ) = 1 . Furthermore, we find fromthe properties of µ ± that s allows analytic extension to C − and s analytically extends to C + .A Riemann-Hilbert problem desired is closely associated with two matrix functions: one isanalytic in C + and the other is analytic in C − . In consideration of the analytic properties of µ ± ,we set P ( x, ς ) = ([ µ + ] , [ µ − ] )( x, ς ) , (9)defining in C + , be an analytic function of ς . And then, P can be expanded into the asymptoticseries at large- ς P = P (0)1 + P (1)1 ς + P (2)1 ς + O (cid:18) ς (cid:19) , ς → ∞ . (10)Inserting expansion (10) into the spectral problem (4a) and equating terms with same powers of ς , we obtain i (cid:2) σ, P (1)1 (cid:3) + U P (0)1 = P (0)1 x , i (cid:2) σ, P (0)1 (cid:3) = 0 , which yields P (0)1 = I , namely P → I as ς ∈ C + → ∞ . For establishing a Riemann-Hilbert problem, the analytic counterpart of P in C − is still neededto be given. Noting that the adjoint scattering equation of (4a) reads as H x = iς [ σ, H ] − HU , (11)and the inverse matrices of µ ± meet this adjoint equation. Then we express the inverse matricesof µ ± as a collection of rows µ − ± = [ µ − ± ] [ µ − ± ] , (12)which obey the boundary conditions µ − ± → I as x → ±∞ . It is easy to know from (8) that E − µ − − = R ( ς ) E − µ − , (13)7here R ( ς ) = ( r kj ) × = S − ( ς ). Thus, the matrix function P which is analytic for ς ∈ C − isconstructed as P ( x, ς ) = [ µ − ] [ µ − − ] ( x, ς ) . (14)Analogous to P , the very large- ς asymptotic behavior of P turns out to be P → I as ς ∈ C − → ∞ . Carrying (5) into Equation (8) gives rise to([ µ − ] , [ µ − ] ) = ([ µ + ] , [ µ + ] ) s s e iςx s e − iςx s , from which we have [ µ − ] = s e iςx [ µ + ] + s [ µ + ] . Hence, P is of the form P = ([ µ + ] , [ µ − ] ) = ([ µ + ] , [ µ + ] ) s e iςx s . On the other hand, via substituting (12) into Equation (13), we get [ µ − − ] [ µ − − ] = r r e iςx r e − iςx r [ µ − ] [ µ − ] , from which we can express [ µ − − ] as[ µ − − ] = r e − iςx [ µ − ] + r [ µ − ] . As a consequence, P is written as P = [ µ − ] [ µ − − ] = r e − iςx r [ µ − ] [ µ − ] . With two matrix functions P and P which are analytic in C + and C − respectively in hand,we are in a position to deduce a Riemann-Hilbert problem for the eighth-order NLS equation (1).After denoting that the limit of P is P + as ς ∈ C + → R and the limit of P is P − as ς ∈ C − → R ,a Riemann-Hilbert problem can be given below P − ( x, ς ) P + ( x, ς ) = s e iςx r e − iςx , (15)8ith its canonical normalization conditions as P ( x, ς ) → I , ς ∈ C + → ∞ ,P ( x, ς ) → I , ς ∈ C − → ∞ , and r s + r s = 1. N -soliton solution Having described a Riemann-Hilbert problem for Equation (1), we now turn to construct its multi-soliton solutions. To achieve the goal, we first need to solve the Riemann-Hilbert problem (15)under the assumption of irregularity, which signifies that both det P and det P possess some zerosin the analytic domains of their own. From the definitions of P and P , we havedet P ( ς ) = s ( ς ) , ς ∈ C + , det P ( ς ) = r ( ς ) , ς ∈ C − , which means that det P and det P have the same zeros as s and r respectively, and r =( S − ) = s .With above analysis, it is now necessary to reveal the characteristic feature of zeros. It can benoticed that the potential matrix Q has the symmetry property Q † = Q, upon which we deduce µ †± ( ς ∗ ) = µ − ± ( ς ) . (16)Here the subscript † stands for the Hermitian of a matrix. In order to facilitate discussion, weintroduce two special matrices H = diag(1 ,
0) and H = diag(0 , , and express (9) and (14) interms of P = µ + H + µ − H , (17a) P = H µ − + H µ − − . (17b)A direct computation of the Hermitian of expression (17a), using the relation (16), generates that P † ( ς ∗ ) = P ( ς ) , ς ∈ C − , (18)and the involution property of scattering matrix S † ( ς ∗ ) = S − ( ς ) , which leads to s ∗ ( ς ∗ ) = r ( ς ) , ς ∈ C − . (19)This equality implies that each zero ± ς k of s results in each zero ± ς ∗ k of r correspondingly.9herefore, our assumption is that det P has simple zeros { ς j ∈ C + , ≤ j ≤ N } and det P hassimple zeros { ˆ ς j ∈ C − , ≤ j ≤ N } , where ˆ ς l = ς ∗ l , ≤ l ≤ N. The full set of the discrete scatteringdata is composed of these zeros and the nonzero column vectors υ j and row vectors ˆ υ j , whichsatisfy the following equations P ( ς j ) υ j = 0 , (20a)ˆ υ j P (ˆ ς j ) = 0 . (20b)Taking the Hermitian of Equation (20a) and using (18) as well as comparing with Equation(20b), we find that the eigenvectors fulfill the relationˆ υ j = υ † j , ≤ j ≤ N. (21)Differentiating Equation (20a) about x and t and taking advantage of the Lax pair (4), we arriveat P ( ς j ) (cid:18) ∂υ j ∂x − iς j συ j (cid:19) = 0 ,P ( ς j ) (cid:18) ∂υ j ∂t − i (cid:0) A ς j + 4 A ς j − A ς j − A ς j + 32 A ς j + 64 A ς j − A ς j (cid:1) συ j (cid:19) = 0 , which yields υ j =e ( iς j x + i (2 A ς j +4 A ς j − A ς j − A ς j +32 A ς j +64 A ς j − A ς j ) t ) σ υ j, , ≤ j ≤ N. Here υ j, , ≤ j ≤ N, are complex constant vectors. Making use of the relation (21), we haveˆ υ j = υ † j, e ( − iς ∗ j x − i (2 A ς ∗ j +4 A ς ∗ j − A ς ∗ j − A ς ∗ j +32 A ς ∗ j +64 A ς ∗ j − A ς ∗ j ) t ) σ , ≤ j ≤ N. However, in order to derive soliton solutions of the eighth-order NLS equation (1), we inves-tigate the Riemann-Hilbert problem (15) corresponding to the reflectionless case, i.e., s = 0.Introducing a N × N matrix M defined as M = ( M kj ) N × N = (cid:18) ˆ υ k υ j ς j − ˆ ς k (cid:19) N × N , ≤ k, j ≤ N, thus the solution to the problem (15) can be determined by P ( ς ) = I − N X k =1 N X j =1 υ k ˆ υ j (cid:0) M − (cid:1) kj ς − ˆ ς j , (22a) P ( ς ) = I + N X k =1 N X j =1 υ k ˆ υ j (cid:0) M − (cid:1) kj ς − ς k , (22b)10here (cid:0) M − (cid:1) kj denotes the ( k, j )-entry of M − . From expression (22a), it can be seen that P (1)1 = − N X k =1 N X j =1 υ k ˆ υ j (cid:0) M − (cid:1) kj . In what follows, we shall retrieve the potential function q ( x, t ) based on the scattering data.Expanding P ( ς ) at large- ς as P ( ς ) = I + P (1)1 ς + P (2)1 ς + O (cid:18) ς (cid:19) , ς → ∞ , and carrying this expansion into (4a) give rise to Q = − (cid:2) σ, P (1)1 (cid:3) . Consequently, the potential function is reconstructed as q ( x, t ) = 2 (cid:0) P (1)1 (cid:1) , with (cid:0) P (1)1 (cid:1) being the (2,1)-entry of P (1)1 .To conclude, setting the nonzero vectors υ k, = ( α k , β k ) T and θ k = iς k x + i (cid:0) A ς k + 4 A ς k − A ς k − A ς k + 32 A ς k + 64 A ς k − A ς k (cid:1) t, Im( ς k ) >
0, the general N -soliton solution for theeighth-order NLS equation (1) is written as q ( x, t ) = − N X k =1 N X j =1 α ∗ j β k e − θ k + θ ∗ j (cid:0) M − (cid:1) kj , (23)where M kj = α ∗ k α j e θ ∗ k + θ j + β ∗ k β j e − θ ∗ k − θ j ς j − ς ∗ k , ≤ k, j ≤ N. The bright one- and two-soliton solutions will be our main concern in the rest of this section.For the simplest case of N = 1, the bright one-soliton solution can be readily derived as q ( x, t ) = − α ∗ β e − θ + θ ∗ ς − ς ∗ | α | e θ ∗ + θ + | β | e − θ ∗ − θ , (24)where θ = iς x + i (cid:0) A ς + 4 A ς − A ς − A ς + 32 A ς + 64 A ς − A ς (cid:1) t . Furthermore,fixing β = 1 and setting ς = ˜ a + i ˜ b as well as | α | = e ξ , the solution (24) is then turned intothe following form q ( x, t ) = − iα ∗ ˜ b e − ξ e θ ∗ − θ sech( θ ∗ + θ + ξ ) , (25)11here θ ∗ + θ = − b (cid:2) x + (cid:0) A ˜ a + 12 A ˜ a − A ˜ a + 32 A ˜ a ˜ b − A ˜ a + 160 A ˜ a ˜ b + 192 A ˜ a − A ˜ a ˜ b + 192 A ˜ a ˜ b + 448 A ˜ a − A ˜ a ˜ b + 1344 A ˜ a ˜ b − A ˜ a − A ˜ b ˜ a + 7168 A ˜ b ˜ a + 1024 A ˜ b ˜ a − A ˜ b − A ˜ b − A ˜ b (cid:1) t (cid:3) ,θ ∗ − θ = − ix ˜ a + 960 itA ˜ a ˜ b + 24 itA ˜ a ˜ b + 896 itA ˜ a ˜ b + 256 itA ˜ a + 16 itA ˜ a + 160 itA ˜ a ˜ b − itA ˜ a + 2688 itA ˜ a ˜ b + 16 itA ˜ b − itA ˜ a + 4 itA ˜ b − itA ˜ a ˜ b − itA ˜ a ˜ b − itA ˜ a ˜ b − itA ˜ a ˜ b + 32 itA ˜ a − itA ˜ a − itA ˜ a ˜ b − itA ˜ a − itA ˜ a ˜ b + 256 itA ˜ b + 64 itA ˜ b + 17920 itA ˜ a ˜ b . Hence we can further write the bright one-soliton solution (25) as q ( x, t ) = − iα ∗ ˜ b e − ξ e θ ∗ − θ sech (cid:8) − b (cid:2) x + (cid:0) A ˜ a + 12 A ˜ a − A ˜ a + 32 A ˜ a ˜ b − A ˜ a + 160 A ˜ a ˜ b + 192 A ˜ a − A ˜ a ˜ b + 192 A ˜ a ˜ b + 448 A ˜ a − A ˜ a ˜ b + 1344 A ˜ a ˜ b − A ˜ a − A ˜ b ˜ a + 7168 A ˜ b ˜ a + 1024 A ˜ b ˜ a − A ˜ b − A ˜ b − A ˜ b (cid:1) t (cid:3) + ξ (cid:9) , (26)from which it is indicated that the solution (26) takes the shape of hyperbolic secant function withpeak amplitude H = 2 | α ∗ | ˜ b e − ξ and velocity V = − A ˜ a − A ˜ a + 32 A ˜ a − A ˜ a ˜ b + 80 A ˜ a − A ˜ a ˜ b − A ˜ a + 640 A ˜ a ˜ b − A ˜ a ˜ b − A ˜ a + 2240 A ˜ a ˜ b − A ˜ a ˜ b + 1024 A ˜ a + 7168 A ˜ b ˜ a − A ˜ b ˜ a − A ˜ b ˜ a + 4 A ˜ b + 16 A ˜ b + 64 A ˜ b . To show the localized structures and dynamic behaviors of one-soliton solution (26), we selectthe involved parameters as ˜ a = 0 . , ˜ b = 0 . , α = A = A = A = A = A = A = A =1 , ξ = 0 . The plots are depicted in Figures 1–3.12hen for the case of N = 2, the bright two-soliton solution for Equation (1) is generated as q ( x, t ) = 2 M M − M M (cid:0) α ∗ β e − θ + θ ∗ M − α ∗ β e − θ + θ ∗ M − α ∗ β e − θ + θ ∗ M + α ∗ β e − θ + θ ∗ M (cid:1) , (27)where M = | α | e θ ∗ + θ + | β | e − θ ∗ − θ ς − ς ∗ , M = α ∗ α e θ ∗ + θ + β ∗ β e − θ ∗ − θ ς − ς ∗ ,M = α ∗ α e θ ∗ + θ + β ∗ β e − θ ∗ − θ ς − ς ∗ , M = | α | e θ ∗ + θ + | β | e − θ ∗ − θ ς − ς ∗ ,θ = iς x + i (cid:0) A ς + 4 A ς − A ς − A ς + 32 A ς + 64 A ς − A ς (cid:1) t,θ = iς x + i (cid:0) A ς + 4 A ς − A ς − A ς + 32 A ς + 64 A ς − A ς (cid:1) t, and ς = ˜ a + i ˜ b , ς = ˜ a + i ˜ b .After assuming that β = β = 1 and α = α as well as | α | = e ξ , the bright two-solitonsolution (27) becomes q ( x, t ) = 2 M M − M M (cid:0) α ∗ e − θ + θ ∗ M − α ∗ e − θ + θ ∗ M − α ∗ e − θ + θ ∗ M + α ∗ e − θ + θ ∗ M (cid:1) , (28)where M = − i ˜ b e ξ cosh( θ ∗ + θ + ξ ) ,M = 2e ξ (˜ a − ˜ a ) + i (˜ b + ˜ b ) cosh( θ ∗ + θ + ξ ) ,M = 2e ξ (˜ a − ˜ a ) + i (˜ b + ˜ b ) cosh( θ ∗ + θ + ξ ) ,M = − i ˜ b e ξ cosh( θ ∗ + θ + ξ ) . The localized structures and dynamic behaviors of two-soliton solution (28) are depicted inFigure 4 via a selection of the parameters as ˜ a = 0 . , ˜ b = 0 . , ˜ b = 0 . , α = α = A = A = A = A = A = A = A = 1 , ˜ a = ξ = ξ = 0 . In this investigation, the aim was to explore multi-soliton solutions for an eighth-order nonlinearSchr¨odinger equation arising in an optical fiber. The method we resort to was the Riemann-Hilbert approach which is based on a Riemann-Hilbert problem. Therefore, we first described aRiemann-Hilbert problem via analyzing the spectral problem of the Lax pair. After solving theobtained Riemann-Hilbert problem corresponding to the reflectionless case, we finally generatedthe expression of general N -soliton solution to the eighth-order nonlinear Schr¨odinger equation. Inaddition, the localized structures and dynamic behaviors of bright one- and two-soliton solutions13 a) (b) Figure 1.
Plots of one-soliton solution (26): (a) Perspective view of modulus of q ; (b) The solitonalong the x -axis with different time in Figure 1(a).(a) (b) Figure 2.
Plots of one-soliton solution (26): (a) Perspective view of real part of q ; (b) The solitonalong the x -axis with different time in Figure 2(a).(a) (b) Figure 3.
Plots of one-soliton solution (26): (a) Perspective view of imaginary part of q ; (b) Thesoliton along the x -axis with different time in Figure 3(a). a) (b) Figure 4.
Plots of two-soliton solution (28): (a) Perspective view of modulus of q ; (b) The solitonalong the x -axis with different time in Figure 4(a). were shown graphically via suitable choices of the involved parameters. Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos.61072147 and 11271008).
References [1] Hu, W.Q., Gao, Y.T., Zhao, C., Feng, Y.J., Su, C.Q.: Oscillations in the interactions amongmultiple solitons in an optical fibre. Z. Naturforsch. , 1079–1091 (2016)[2] Hu, W.Q., Gao, Y.T., Zhao, C., Lan, Z.Z.: Breathers and rogue waves for an eighth-ordernonlinear Schr¨odinger equation in an optical fiber. Mod. Phys. Lett. B , 1750035 (2017)[3] Ankiewicz, A., Kedziora, D.J., Chowdury, A., Bandelow, U., Akhmediev, N.: Infinite hierarchyof nonlinear Schr¨odinger equations and their solutions. Phys. Rev. E , 012206 (2016)[4] Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. ,805–809 (1973)[5] Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of theHirota equation. Phys. Rev. E , 046602 (2010)[6] Porsezian, K., Daniel, M., Lakshmanan, M.: On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain. J. Math. Phys. , 1807–1816 (1992) 157] Yang, B., Zhang, W.G., Zhang, H.Q., Pei S.B.: Generalized Darboux transformation androgue wave solutions for the higher-order dispersive nonlinear Schr¨odinger equation. Phys.Scr. , 065004 (2013)[8] Chai, J., Tian, B., Zhen, H.L., Sun, W.R.: Conservation laws, bilinear forms and solitons fora fifth-order nonlinear Schr¨odinger equation for the attosecond pulses in an optical fiber. Ann.Physics , 371–384 (2015)[9] Zhang, Y.S., Cheng, Y., He, J.S.: Riemann-Hilbert method and N -soliton for two-componentGerdjikov-Ivanov equation. J. Nonlinear Math. Phys. , 210–223 (2017)[10] Boutet de Monvel, A., Shepelsky, D.; A Riemann-Hilbert approach for the Degasperis-Procesiequation. Nonlinearity , 2081–2107 (2013)[11] Boutet de Monvel, A., Shepelsky, D.; The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach. J. Phys. A: Math. Theor. , 035204 (2015)[12] Zhang, N., Hu, B.B., Xia, T.C.: A Riemann-Hilbert approach to complex Sharma-Tasso-Olverequation on half line. Commun. Theor. Phys. , 580–594 (2017)[13] Ma, W.X., Dong, H.H.: Modeling Riemann-Hilbert problems to get soliton solutions. Math.Model. Appl. , 16–25 (2017)[14] Hu, B.B., Xia, T.C., Zhang, N., Wang, J.B.: Initial-boundary value problems for the coupledhigher-order nonlinear Schr¨odinger equations on the half-line. Int. J. Nonlinear Sci. Numer.Simul. , 83–92 (2018)[15] Hu, B.B., Xia, T.C., Ma, W.X.: Riemann-Hilbert approach for an initial-boundary valueproblem of the two-component modified Korteweg-de Vries equation on the half-line. App.Math. Comput. , 148–159 (2018)[16] Hu, B.B., Xia, T.C., Ma, W.X.: The Riemann-Hilbert approach to initial-boundary valueproblems for integrable coherently coupled nonlinear Schr¨odinger systems on the half-line.East Asian J. Appl. Math. , 531–548 (2018)[17] Ma, W.X.: Riemann-Hilbert problems and N -soliton solutions for a coupled mKdV system.J. Geom. Phys. , 45–54 (2018)[18] Ma, W.X.: Riemann-Hilbert problems of a six-component fourth-order AKNS system and itssoliton solutions. Comput. Appl. Math. (2018) https://doi.org/10.1007/s40314-018-0703-6[19] Guo, B.L., Liu, N., Wang, Y.F.: A Riemann-Hilbert approach for a new type coupled nonlinearSchr¨odinger equations. J. Math. Anal. Appl. , 145–158 (2018)[20] Kang, Z.Z., Xia, T.C., Ma, X.: Multi-soliton solutions for the coupled modified nonlinearSchr¨odinger equations via Riemann-Hilbert approach. Chin. Phys. B27