Rogue waves in the generalized derivative nonlinear Schrodinger equations

Rogue w a v es in the generalized deriv ativ e nonlinear Sc hr¨ odinger equations Bo Y ang a , Junc hao Chen b and Jiank e Y ang a a Dep artment of Mathematics and Statistics, University of V ermont, Burlington, VT 05405, U.S.A. b Dep artment of Mathematics, Lishui University, Lishui 323000, China General rogue w a ves are deriv ed for the generalized deriv ativ e nonlinear Sc hr¨ odinger (GDNLS) equations b y a bilinear Kadomtsev-P etviash vili (KP) reduction method. These GDNLS equations con tain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjiko v-Iv ano v equation as sp ecial cases. In this bilinear framework, it is shown that rogue wa ves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction metho ds for rogue wa ves in other integrable equations, an imp ortant improv emen t in our current KP reduction pro cedure is a new parameterization of internal parameters in rogue w av es. Under this new parameterization, the rogue wa ve expressions through elementary Sch ur p olynomials are m uch simpler. In addition, the rogue wa v e with the highest p eak amplitude at each order can b e obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order N turns out to b e 2 N + 1 times the background amplitude, indep endent of the individual GDNLS equation and the background w a ven um b er. It is also rep orted that these GDNLS equations can b e decomp osed in to t wo different bilinear systems whic h require different KP reductions, but the resulting rogue wa ves remain the same. Dynamics of rogue wa v es in the GDNLS equations is also analyzed. It is shown that the wa ven um b er of the constant background strongly affects the orien tation and duration of the rogue wa ve. In addition, some new rogue patterns are presen ted. I. INTR ODUCTION Rogue wa ves are large and sp ontaneous lo cal excitations of nonlinear wa v e equations that “app ear from nowhere and disappear with no trace” [1]. More specifically , these lo cal excitations arise from a flat constan t-amplitude bac kground, reach a transient high amplitude, and then retreat back to the same flat background. Suc h solutions w ere first rep orted for the nonlinear Schr¨ odinger (NLS) equation by P eregrine in 1983 [2]. In recent years, such wa ves w ere link ed to freak w av es in the o cean [3, 4] and extreme ev ents in optics [5, 6], and w ere observ ed in water-tank and optical-fib er exp erimen ts [7 – 11]. Motiv ated b y these physical applications, rogue wa v es hav e b een derived in a large n umber of physically-relev ant in tegrable nonlinear wa v e equations, including the NLS equation [12 – 18], the deriv ativ e NLS equations [19 – 22], the Manako v equations [23, 24], the Da vey-Stew artson equations [25, 26], and many others [27 – 36]. Indeed, rogue wa ves are caused by baseband mo dulation instability of the constant-amplitude background [24]. Th us, any in tegrable equation with baseband mo dulation instability is exp ected to admit rogue w av es, which can b e deriv ed by in tegrable techniques. All known rogue w av es in in tegrable equations are rational solutions of the underlying systems. This fact is related to baseband mo dulation instability , since rational rogue-wa v e solutions are asso ciated with long-wa v e instability of the background. W e note by passing that in nonintegrable systems, large and sp on taneous lo cal excitations can also arise from a constan t-amplitude background if suc h background admits baseband mo dulation instability (see [5] for instance). But such excitations do not retreat back to the same background, and are not exp ected to admit exact analytical expressions, due to the lack of integrabilit y of the underlying nonlinear w av e equations [37]. In this pap er, we consider rogue wa v es in the generalized deriv ative nonlinear Sc hr¨ odinger (GDNLS) equations [38, 39] i φ t + φ ξξ + ρ | φ | 2 φ + i aφφ ∗ φ ξ + i bφ 2 φ ∗ ξ + 1 4 b (2 b − a ) | φ | 4 φ = 0 , (1) where ρ, a, b are arbitrary real constants with a 6 = b , and the sup erscript ‘*’ represents complex conjugation (the a = b case will b e treated in the app endix). In fib er optics, these equations mo del the propagation of short light pulses where, in addition to disp ersion and Kerr (cubic) nonlinearity , self-steep ening and fifth-order nonlinearity are also accounted for (even though the Raman effect and third-order disp ersion are omitted) [40, 41]. When ρ = 0 and b = 2 a , these equations reduce to the Kaup-New ell equation [42], whic h go v erns the propagation of circularly polarized nonlinear Alfv´ en wa ves in plasmas [43, 44]. When ρ = b = 0, these equations reduce to the Chen-Lee-Liu equation [45], which mo dels short-pulse propagation in a frequency-doubling crystal through the interpla y of quadratic and cubic nonlinearities [46]. Due to these physical applications, rogue wa v e formation in these GDNLS equations is a ph ysically significan t issue. There hav e b een a num b er of studies on rogue w av es in these GDNLS equations. F or instance, for the Kaup- New ell equation (with ρ = 0 and b = 2 a ), special t yp es of rogue w av es w ere derived b y Darb oux transformation in [19, 20]. F or the Chen-Lee-Liu-type equation, with b = 0 in (1), the fundamen tal rogue w av e was derived b y the 2 bilinear Hirota metho d in [21], and higher-order rogue wa ves were derived by Darb oux transformation in [22]. F or the Gerdjiko v-Iv anov equation [47], with ρ = a = 0 in (1), fundamental and higher-order rogue wa v es w ere deriv ed b y Darb oux transformation in [48, 49]. Even for the GDNLS equations (1) themselves, general rogue wa ves were derived b y Darb oux transformation in [50], and their chirping phase structure w as examined. In this article, we deriv e general rogue w av es in the GDNLS equations (1) by the bilinear Kadomtsev-P etviash vili (KP) reduction method. The adv antage of this bilinear framew ork is that rogue wa v es in all GDNLS equations (1) can b e expressed explicitly by the same bilinear solution. Compared to previous bilinear KP reduction metho ds for rogue w av es in other integrable equations [18, 25 – 30], an imp ortant improv emen t in our current KP reduction technique is a new parameterization of internal parameters in rogue wa ves. Under this parameterization, analytical expressions of rogue wa v es through Sch ur p olynomials are muc h simpler. More imp ortan tly , when all internal parameters are set to zero, we w ould get a parity-time-symmetric rogue wa v e which attains the maximum p eak amplitude among rogue wa v es of that order. This allo ws us to analytically derive this maximum p eak amplitude, whic h turns out to b e 2 N + 1 times the background amplitude at order N , indep enden t of the individual GDNLS equation and the bac kground wa ven umber. W e also find that the GDNLS equations (1) can be decomp osed into tw o different bilinear systems which require differen t KP reductions, but the resulting rogue w av es are the same. After these rogue wa ves are derived, their dynamics is also analyzed. It is shown that the wa v en umber of the bac kground strongly affects the orien tation and duration of the rogue wa v e. In addition, some new rogue patterns are presented. In the app endix, general rogue w a v es for the GDNLS equations (1) with a = b (the so-called Kundu-Ec khaus equation) are also given in the bilinear framew ork. These results deep en our understanding of rogue w av es in the physically significan t GDNLS equations (1). Meanwhile, they adv ance the bilinear KP-reduction methodology for the deriv ation of rogue w av es. I I. PRELIMINARIES Under a simple gauge transformation [51] φ ( ξ , t ) = r 2 a − b u ( x, t ) exp  i ρ a − b x + i ρ 2 ( a − b ) 2 t  , where x = ξ − 2 ρt/ ( a − b ), the GDNLS equations (1) with a 6 = b reduce to i u t + u xx + 2i γ | u | 2 u x + 2i( γ − 1) u 2 u ∗ x + ( γ − 1)( γ − 2) | u | 4 u = 0 , (2) where γ = a/ ( a − b ). W e will work with these normalized GDNLS equations (2) in the remainder of this article. These equations become the Kaup-Newell equation when γ = 2 [42], the Chen-Lee-Liu equation when γ = 1 [45], and the Gerdjik o v-Iv ano v equation when γ = 0 [47]. It is noted that with an additional gauge transformation u ( x, t ) = v ( x, t ) e i(2 − γ ) R | v ( x,t ) | 2 dx , (3) the normalized GDNLS equations (2) further reduce to the Kaup-New ell equation i v t + v xx + 2i( | v | 2 v ) x = 0 . (4) Th us, from rogue wa v es of the Kaup-New ell equation, one can deriv e rogue w a ves in the GDNLS equations (2) in principle. Ho wev er, the gauge transformation (3) inv olv es a non trivial integral, which makes it difficult to deriv e explicit solutions to the GDNLS equations from those of the Kaup-New ell equation. F or this reason, we will not utilize this gauge transformation. Instead, we will use a bilinear metho d to directly obtain explicit rogue wa v e solutions in the GDNLS equations (2) for arbitrary γ v alues. Rogue wa ves in the GDNLS equations (2) approach a constant-amplitude con tinuous wa ve background at large x and t . By simple v ariable scalings, this constant amplitude can b e normalized to b e unit y . Then, these rogue w av es approac h the unit-amplitude contin uous w av e bac kground e i κx − i ω t , where κ is a free wa v enum ber, and ω = κ 2 + 2 κ − ( γ − 1)( γ − 2) is the frequency . In order for rogue wa ves to arise, these backgrounds must be unstable to baseband mo dulations [24]. Simple mo dulation instabilit y calculations sho w that these backgrounds are base-band unstable when κ < 1 − γ . Thus, rogue wa v es in the GDNLS equations (2) should approach the following bac kground as x, t → ±∞ : u ( x, t ) → e i(1 − γ − α ) x − i [ α 2 +2( γ − 2) α +1 − γ ] t , (5) 3 where α > 0 is a wa ven umber parameter. Unlike the NLS equation, the GDNLS equations (2) do not admit Galilean- transformation in v ariance. Thus, α is a non-reducible parameter in its rogue wa ves. In view of the ab o ve b oundary condition, we introduce the v ariable transformation u = e i(1 − γ − α ) x − i [ α 2 +2( γ − 2) α +1 − γ ] t ( f ∗ ) γ − 1 g f γ , (6) where f and g are complex functions. Under this transformation, the GDNLS equations (2) can b e decomp osed into the follo wing system of four bilinear equations:  i D t + D 2 x + 2i(1 − α ) D x  g · f ∗ = 0 , (7)  i D t + D 2 x + 2i D x  f · f ∗ = 0 , (8) (i D x − 1) f · f ∗ + | g | 2 = 0 , (9) D 2 x f · f ∗ − i D x g · g ∗ + (2 α + 1)( | f | 2 − | g | 2 ) = 0 , (10) where D is Hirota’s bilinear differential op erator. W e will use these bilinear equations to deriv e rogue w av es in the GDNLS equations (2). It is imp ortant to notice that these bilinear equations are indep endent of the equation parameter γ . This means that rogue wa v es in the whole family of GDNLS equations (2), for different v alues of γ , are given by the same f and g solutions, and the γ -dep endence of the rogue w av es only appears through the bilinear transformation (6). This is a big adv an tage of the bilinear metho d for solving the GDNLS equations (2). In terestingly , under the same transformation (6), the GDNLS equation (2) can also b e decomp osed into a different bilinear system, where the first equation (7) is replaced b y  i D t + D 2 x − 2i αD x  g · f = 0 , (11) while the other three equations (8)-(10) remain the same. This replacemen t is admitted b ecause the left side of the latter first bilinear equation (11) can b e written as a linear combination of the left sides of the former bilinear equations (7)-(10). Sp ecifically , denoting the left side of each equation by its equation num b er, we hav e the identit y f × (7) − g × (8) + 2(i g x + αg ) × (9) = f ∗ × (11) − g × (10) + g (i ∂ x − 1) × (9) . (12) Th us, if f and g satisfy the former system of bilinear equations, then they would also satisfy the latter bilinear system. Although these tw o (1 + 1)-dimensional bilinear systems are equiv alent, they ha ve to be reduced from differen t higher- dimensional bilinear systems whic h admit differen t bilinear solutions. But these tw o differen t KP reductions will lead to the same rogue wa ve solutions, which we will show in later texts. In this article, we will present rogue wa ves of the GDNLS equations (2) through elementary Sch ur p olynomials. These Sc h ur p olynomials S j ( x ) are defined by ∞ X j =0 S j ( x ) λ j = exp   ∞ X j =1 x j λ j   , or more explicitly , S 0 ( x ) = 1 , S 1 ( x ) = x 1 , S 2 ( x ) = 1 2 x 2 1 + x 2 , · · · , S j ( x ) = X l 1 +2 l 2 + ··· + ml m = j   m Y j =1 x l j j l j !   , where x = ( x 1 , x 2 , · · · ). I II. GENERAL ROGUE W A VE SOLUTIONS Our general rogue wa v e solutions to the GDNLS equations (2) are given by the follo wing theorem. Theorem 1. The GDNLS e quations (2) under the b oundary c ondition (5) admit r ational r o gue wave solutions u N ( x, t ) = e i(1 − γ − α ) x − i [ α 2 +2( γ − 2) α +1 − γ ] t ( f ∗ N ) γ − 1 g N f γ N , (13) 4 wher e the p ositive inte ger N r epr esents the or der of the r o gue wave, f N ( x, t ) = σ 0 , 0 , g N ( x, t ) = σ − 1 , 1 , σ n,k = det 1 ≤ i,j ≤ N  m ( n,k ) 2 i − 1 , 2 j − 1  , the matrix elements in σ n,k ar e define d by m ( n,k ) i,j = min( i,j ) X ν =0 1 4 ν S i − ν ( x + ( n, k ) + ν s ) S j − ν ( x − ( n, k ) + ν s ) , (14) ve ctors x ± ( n, k ) =  x ± 1 , x ± 2 , · · ·  ar e define d by x + 1 = k +  n + 1 2   h 1 + 1 2  + √ αx + 2 √ α  ( α − 1) + i √ α  t + a 1 , x − 1 = − k −  n + 1 2   h ∗ 1 + 1 2  + √ αx + 2 √ α  ( α − 1) − i √ α  t + a ∗ 1 , x + r = ( n + 1 2 ) h r + 1 r !  √ αx +  2 √ α ( α − 1) + 2 r i α  t  + a r , r > 1 , x − r = − ( n + 1 2 ) h ∗ r + 1 r !  √ αx +  2 √ α ( α − 1) − 2 r i α  t  + a ∗ r , r > 1 , h r ( α ) , s r ar e c o efficients fr om the exp ansions ∞ X r =1 h r λ r = ln  i e λ/ 2 + √ αe − λ/ 2 i + √ α  , ∞ X r =1 s r λ r = ln  2 λ tanh  λ 2  , (15) and a r ( r = 1 , 2 , . . . ) ar e fr e e c omplex c onstants. Note 1. The first few co efficients h r ( α ) and s r in expansions (15) are h 1 ( α ) = i − √ α 2 (i + √ α ) , h 2 ( α ) = i √ α 2 (i + √ α ) 2 , h 3 ( α ) = √ α (1 + i √ α ) 6 (i + √ α ) 3 , (16) s 1 = s 3 = · · · = s odd = 0 , s 2 = − 1 12 , s 4 = 7 1440 . (17) Theorem 1 will b e pro ved in Sec. V. Some remarks on rogue wa ves in this theorem are in order. First, one can notice that the matrix-element expression in this theorem is significantly simpler than earlier suc h expressions for other in tegrable equations [18, 25 – 28]. Indeed, the curren t expression in (14) in v olv es a single summation, while previous suc h expressions in volv ed three summations. Second, our curren t parameterization of rogue w a v es in Theorem 1 is very different from the previous ones. In our curren t rogue wa v e solution, all internal parameters a 1 , a 2 , a 3 , . . . appear inside the x ± ( n, k ) vectors, while previous in ternal parameters all app eared outside such vectors as summation co efficients [18, 25–28]. This different parameterization is the key reason for the simpler matrix-elemen t expression in Theorem 1. More significan tly , this parameterization facilitates the analysis of rogue wa ves, esp ecially regarding the maximum p eak amplitude for rogue w av es of a given order. Indeed, under previous parameterizations for the NLS equation, the rogue wa ve with maxim um p eak amplitude o ccurs at peculiar in ternal parameter v alues [18], which mak es the deriv ation of maximum p eak amplitudes at arbitrary orders intractable. Ho w ev er, in our current parameterization, rogue w av es in Theorem 1 admit the following prop ert y . Theorem 2. When a r = 0 for al l r ≥ 1 , the r o gue wave in The or em 1 is p arity-time-symmetric, i.e., u ∗ N ( − x, − t ) = u N ( x, t ) . This prop ert y will also be prov ed in Sec. V. The significance of this property is that, this parit y-time-symmetric rogue w av e happ ens to possess the maximum p eak amplitude among rogue wa ves of that order (see [50]). In addition, this maxim um p eak amplitude is lo cated at the center of this parity-time-symmetric rogue wa v e, i.e., at x = t = 0. Thus, 5 to deriv e the maximum p eak amplitude of rogue wa ves in Theorem 1, we only need to set all its internal parameters a r as w ell as ( x, t ) to zero, whic h is muc h easier. Doing so, our explicit calculations for N = 1 , 2 , . . . , 6 sho w that | f N (0 , 0) | a r =0 = α N ( N +1) / 2 2 2 N 2 ( α + 1) N ( N +1) / 2 , | g N (0 , 0) | a r =0 = (2 N + 1) α N ( N +1) / 2 2 2 N 2 ( α + 1) N ( N +1) / 2 , (18) and th us the maximum p eak amplitude is | u N (0 , 0) | a r =0 = | g N (0 , 0) | a r =0 | f N (0 , 0) | a r =0 = 2 N + 1 . (19) Remark ably , this maxim um p eak amplitude do es not dep end on the bac kground wa v en umber α , although | f N | and | g N | in its numerator and denominator do. While these formulae (18)-(19) were obtained for N ≤ 6, w e b eliev e they hold for all N > 6 as well. In Refs. [22, 49, 50] for the Chen-Lee-Liu equation, the Gerdjiko v-Iv anov equation and the GDNLS equations (1), examination of some low-order rogue wa ves revealed that their maxim um p eak amplitude w as 2 N + 1. Our result ab o v e is more general. Interestingly , this maximum p eak amplitude for rogue wa ves in the GDNLS equations (2) is exactly the same as that for the NLS equation [12, 13, 18, 52]. Another remark on rogue wa v es in Theorem 1 p ertains to the num b er of their irreducible free parameters. These rogue wa v es of order N contain 2 N − 1 complex parameters a 1 , a 2 , . . . , a 2 N − 1 . But w e can show that all ev en-indexed parameters a ev en are dummy parameters whic h cancel out automatically from the solution. T o prov e this, we first rewrite σ n,k in Theorem 1 as [18] σ n,k = X 0 ≤ ν 1 <ν 2 < ··· <ν N ≤ 2 N − 1 det 1 ≤ i,j ≤ N  1 2 ν j S 2 i − 1 − ν j ( x + ( n, k ) + ν j s )  det 1 ≤ i,j ≤ N  1 2 ν j S 2 i − 1 − ν j ( x − ( n, k ) + ν j s )  . (20) In addition, denoting ξ r and η r as the real and imaginary parts of a r , w e can easily see that ∂ ξ r S n ( x ± + ν s ) = S n − r ( x ± + ν s ) , ∂ η r S n ( x ± + ν s ) = ± i S n − r ( x ± + ν s ) . (21) Using these tw o equations, we can show that ∂ ξ 2 r σ n,k = ∂ η 2 r σ n,k = 0 , (22) whic h prov es that rogue wa ves in Theorem 1 are indep endent of parameters a ev en . Th us, we will simply set a 2 = a 4 = · · · = a ev en = 0 throughout this article. Of the remaining parameters, we can normalize a 1 = 0 through a shift of x and t . Then, the N -th order rogue wa v es in the GDNLS equation (2) contain N − 1 free irreducible complex parameters, a 3 , a 5 , . . . , a 2 N − 1 . This num ber of irreducible free parameters is the same as that in rogue wa v es of the NLS equation [18]. IV. D YNAMICAL P A TTERNS OF ROGUE W A VES In this section, we analyze the dynamics of rogue wa v es in Theorem 1 for the GDNLS equations (2). First of all, we notice from Eq. (13) that the amplitude profile of the rogue wa ve is | u N ( x, t ) | = | g N ( x, t ) | | f N ( x, t ) | , (23) whic h is indep enden t of the equation parameter γ . This means that the in tensit y patterns of rogue w a ves are the same for al l GDNLS equations (2) regardless of the γ v alue. But the phase structure of rogue wa v es is influenced by the γ v alue. Indeed, the gauge transformation (3) tells us that, on top of rogue w av es v ( x, t ) of the Kaup-Newell equation, differen t v alues of γ introduce an extra phase θ ( x, t ) = (2 − γ ) R | v ( x, t ) | 2 dx , which can b e calculated directly from the bilinear solution (13). This phase induces a “chirp” to an optical rogue wa v e, whic h was examined in detail in [50]. Although the rogue wa v e intensit y pattern in the GDNLS equations (2) is independent of γ , it does dep end on the w av en um b er parameter α of the constant bac kground. W e will fo cus on this α dep endence of the rogue-wa v e intensit y pattern next. First, we consider fundamen tal rogue wa ves, where we set N = 1 in Theorem 1. In addition, we normalize a 1 = 0 (see the remark in the end of the last section). Then, we get | u 1 ( x, t ; α ) | =     α ( x + 2 αt ) 2 + ( x − 2 t ) 2 − i( x + 6 αt ) − 3 4 α ( x + 2 αt ) 2 + ( x − 2 t ) 2 + i( x + 2 αt ) − 4i t + 1 4     . (24) 6 A t three v alues of α = 0 . 5 , 1 and 2, this amplitude profile is shown in Fig. 1(a,b,c) resp ectiv ely . It is seen that α strongly affects the orien tation and duration of the rogue wa v e. Sp ecifically , as the α v alue increases, the orientation angle also increases, but the duration of the rogue wa v e decreases. How ev er, the p eak amplitudes of these rogue wa v es for different α v alues are all equal to 3, which are attained at the center x = t = 0, i.e., | u 1 (0 , 0; α ) | = 3. Physically , the longer duration of rogue wa v es at smaller α v alues can b e understo od, b ecause in this case, the gro wth rates of baseband mo dulation instability can be shown to b e smaller, whic h causes the rogue wa v e to tak e longer time to rise from the unit-amplitude background to its p eak amplitude of 3. The dep endence of the orientation angle on α can also b e heuristically understo o d. It is known that the phase gradien t of a pulse generally causes the pulse to mo ve at a velocity whic h is prop ortional to this phase gradient. In the present case, the phase gradient of the rogue wa v e can b e estimated from Eq. (13) as the wa v en umber 1 − γ − α . Then, for a fixed equation parameter γ , larger α causes the v elo cit y to b e smaller or negative, leading to a larger orientation angle. T o put these results in p ersp ective, we note that for the NLS equation, since the constant-bac kground wa v enum ber of its rogue wa v es can b e normalized by a Galilean transformation [18], the bac kground wa ven um ber only affects the orientation, but not duration, of its rogue w av es. FIG. 1: Amplitude profiles (24) of first-order rogue wa ves. (a) α = 0 . 5; (b) α = 1; (c) α = 2. It is interesting to notice that, in the limit of α → 0, | u 1 ( x, t ; α ) | →     ( x − 2 t ) 2 − i x − 3 4 ( x − 2 t ) 2 + i x − 4i t + 1 4     , (25) whic h becomes a quadratic algebraic soliton instead of a rogue w a ve. This comes ab out b ecause when α = 0, baseband mo dulation instability disapp ears in the GDNLS equation (2), and thus rogue wa v es no longer exist. No w we consider second-order rogue wa v es, where we set N = 2 and a 1 = 0 in Theorem 1. These solutions contain one complex free parameter a 3 . When a 3 = 0, the resulting rogue wa v e is parity-time-symmetric, and it reaches p eak amplitude 5 at the center x = t = 0 for all α v alues, i.e., | u 2 (0 , 0; α ) | = 5. This p eak amplitude 5 is the maxim um p eak amplitude for all rogue w av es of second order, and thus this parity-time-symmetric rogue wa v e was called the sup er rogue wa ve in [50]. The amplitude profile of this sup er rogue wa ve dep ends on the w av en umber parameter α though. At three α v alues of 0.5, 1 and 2, these sup er rogue w av es are display ed in Fig. 2. Again, α strongly affects the orien tation and duration of these rogue wa ves. When a 3 6 = 0, the second-order rogue wa v es generally will split into three separate first-order rogue wa ves, as has b een reported in [20, 22, 49, 50]. This phenomenon is similar to second-order rogue w av es of the NLS equation [12–18]. The orientations and durations of these three separate first-order rogue wa v es are determined by the wa ven um ber parameter α . Ha ving clarified the effect of w av en um b er parameter α on rogue w av es, at third order, we will fix α = 1 and explore new rogue wa ve patterns. F or this purp ose, we set N = 3 and a 1 = 0, and the remaining free complex parameters are a 3 and a 5 . When a 3 = a 5 = 0, w e get a sup er rogue wa v e with p eak amplitude 7 (see also [22, 49, 50]). At other a 3 and a 5 v alues, the third-order rogue wa ve generally splits into 6 separate first-order rogue wa v es in v arious configurations. Tw o such solutions are display ed in Fig. 3. The left panel shows a p entagon pattern, which has b een seen b efore [22, 49]. But the right panel shows a mix of a first-order rogue wa v e and a cluster of five first-order rogue wa ves in square configuration, which is nov el to our kno wledge. Our results suggest that when a third-order rogue wa v e splits in to 6 separate first-order rogue wa v es, these 6 first-order rogue w av es can app ear in arbitrary configurations in the ( x, t ) plane. The same should hold for higher-order rogue wa ves to o. 7 FIG. 2: Amplitude profiles | u 2 ( x, t ) | of second-order sup er rogue wa v es (with a 3 = 0). (a) α = 0 . 5; (b) α = 1; (c) α = 2. FIG. 3: Third-order rogue w av es with α = 1. Left: a p en tagon pattern, where a 3 = 0 and a 5 = 80 + 80i. Right: a mixed pattern, where a 3 = 10i and a 5 = 100i. V. DERIV A TION OF ROGUE W A VES FROM THE FIRST BILINEAR SYSTEM As w e ha ve mentioned in Sec. I I, the GDNLS equation (2) can b e decomp osed into tw o different bilinear systems. In this section, we will derive rogue wa v es in Theorem 1 from the first bilinear system (7)-(10). The basic idea of this deriv ation is similar to that in [18] for the NLS equation. The main improv emen t is that we will choose differential op erators in the bilinear solutions in a different wa y , whic h leads to a more conv enien t parameterization and simpler expression for rogue wa v es. A. Gram determinant solutions for a higher-dimensional bilinear system First, we need to derive algebraic solutions to a higher-dimensional bilinear system, whic h can reduce to the original lo wer-dimensional bilinear system (7)-(10) under certain reductions. F rom Lemma 2 of Ref. [53], section 3.2 of Ref. [54] and our own calculations, we learn that if functions m ( n,k ) i,j , ϕ ( n,k ) i and ψ ( n,k ) j of v ariables ( x − 1 , x 1 , x 2 ) satisfy the following differential and difference relations, ∂ x 1 m ( n,k ) i,j = ϕ ( n,k ) i ψ ( n,k ) j , ∂ x 1 ϕ ( n,k ) i = ϕ ( n +1 ,k ) i , ∂ x 1 ψ ( n,k ) j = − ψ ( n − 1 ,k ) j , ∂ x 1 ϕ ( n,k ) i = cϕ ( n,k ) i + ϕ ( n,k +1) i , ∂ x 1 ψ ( n,k ) j = − cψ ( n,k ) j − ψ ( n,k − 1) j , ∂ x 2 ϕ ( n,k ) i = ∂ 2 x 1 ϕ ( n,k ) i , ∂ x 2 ψ ( n,k ) j = − ∂ 2 x 1 ψ ( n,k ) j , ∂ x − 1 ϕ ( n,k ) i = ϕ ( n,k − 1) i , ∂ x − 1 ψ ( n,k ) j = − ψ ( n,k +1) j , (26) 8 where c is an arbitrary complex constant, then they would also satisfy the following relations, ∂ x 2 m ( n,k ) i,j = ϕ ( n +1 ,k ) i ψ ( n,k ) j + ϕ ( n,k ) i ψ ( n − 1 ,k ) j , ∂ x 2 m ( n,k ) i,j = ϕ ( n,k +1) i ψ ( n,k ) j + ϕ ( n,k ) i ψ ( n,k − 1) j + 2 cϕ ( n,k ) i ψ ( n,k ) j , ∂ x − 1 m ( n,k ) i,j = − ϕ ( n,k − 1) i ψ ( n,k +1) j , m ( n +1 ,k ) i,j = m ( n,k ) i,j + ϕ ( n,k ) i ψ ( n +1 ,k ) j , m ( n,k +1) i,j = m ( n,k ) i,j + ϕ ( n,k ) i ψ ( n,k +1) j . (27) Using these relations, one can sho w that the determinant τ n,k = det 1 ≤ i,j ≤ N  m ( n,k ) i,j  (28) w ould satisfy the following bilinear equations in the extended KP hierarch y  D x 2 − D 2 x 1 − 2 cD x 1  τ n − 1 ,k +1 · τ n − 1 ,k = 0 , (29)  D x 2 − D 2 x 1  τ n,k · τ n − 1 ,k = 0 , (30)  cD x − 1 − 1  τ n,k · τ n − 1 ,k + τ n − 1 ,k +1 τ n,k − 1 = 0 , (31) ( cD x 1 D x − 1 − D x 1 − 2 c ) τ n,k · τ n − 1 ,k + ( D x 1 + 2 c ) τ n − 1 ,k +1 · τ n,k − 1 = 0 . (32) No w, w e introduce functions m ( n,k ) , ϕ ( n,k ) and ψ ( n,k ) as m ( n,k ) = i p p + q  − p q  n  − p − c q + c  k e ξ + η , (33) ϕ ( n,k ) = (i p ) p n ( p − c ) k e ξ , (34) ψ ( n,k ) = ( − q ) − n [ − ( q + c )] − k e η , (35) where ξ = 1 p − c x − 1 + px 1 + p 2 x 2 + ξ 0 , (36) η = 1 q + c x − 1 + q x 1 − q 2 x 2 + η 0 , (37) and p, q , ξ 0 and η 0 are arbitrary complex constants. It is easy to see that these functions satisfy the differen tial and difference relations (26) with indices i and j ignored. Then, b y defining m ( n,k ) ij = A i B j m ( n,k ) , ϕ ( n,k ) i = A i ϕ ( n,k ) , ψ ( n,k ) j = B j ψ ( n,k ) , (38) where A i and B j are differen tial operators with resp ect to p and q resp ectiv ely as A i = 1 i ! [( p − c ) ∂ p ] i , B j = 1 j ! [( q + c ) ∂ q ] j , (39) these functions would also satisfy the differen tial and difference relations (26) since op erators A i and B j comm ute with differen tials ∂ x k . Consequently , for an arbitrary sequence of indices ( i 1 , i 2 , · · · , i N ; j 1 , j 2 , · · · , j N ), the determinant τ n,k = det 1 ≤ ν,µ ≤ N  m ( n,k ) i ν ,j µ  (40) satisfies the higher-dimensional bilinear system (29)-(32). It is imp ortant to notice that the differen tial op erators A i and B j defined here are simpler than the ones in previous bilinear deriv ations of rogue w a v es [18, 25 – 30]. Indeed, the current differential op erators are single terms, while previous ones were defined as summations. The reason for the previous summation definitions w as to introduce in ternal free parameters in rogue wa ves. In our curren t approac h, w e will introduce free constan ts through ξ 0 and η 0 in Eqs. (36)-(37), which will be done later in this section. Next, w e will reduce the higher-dimensional bilinear system (29)-(32) to the original bilinear system (7)-(10), so that the higher-dimensional solutions (40) b ecome rogue wa v e solutions to the GDNLS equations (2). In this reduction, w e will need to set c = − i α, (41) where c is the parameter in the higher-dimensional system (29)-(32), and α is the wa v en umber parameter in the original bilinear system (7)-(10). 9 B. Dimensional reduction First, we reduce the higher-dimensional bilinear system (29)-(32) to a lo wer-dimensional one, a pro cess called dimension reduction. This reduction will restrict the indices in the determinant (40), and select the ( p, q ) v alues in its matrix element m ( n,k ) i ν ,j µ . The dimension reduction condition we imp ose is  ∂ x 1 + i c∂ x − 1  τ n,k = C τ n,k , (42) where C is some constant. Denoting ˆ p ≡ p − c and ˆ q ≡ q + c , then A i and B j in Eq. (39) can b e rewritten as A i = 1 i ! ( ˆ p∂ ˆ p ) i , B j = 1 j ! ( ˆ q ∂ ˆ q ) j . (43) In addition,  ∂ x 1 + i c∂ x − 1  m ( n,k ) i,j = A i B j  ∂ x 1 + i c∂ x − 1  m ( n,k ) = A i B j  ˆ p + i c ˆ p + ˆ q + i c ˆ q  m ( n,k ) . Using the Leibnitz rule exactly as in Ref. [18], the abov e equation reduces to  ∂ x 1 + i c∂ x − 1  m ( n,k ) i,j = i X µ =0 1 µ !  ˆ p + ( − 1) µ i c ˆ p  m ( n,k ) i − µ,j + j X l =0 1 l !  ˆ q + ( − 1) l i c ˆ q  m ( n,k ) i,j − l . Recalling c = − i α from (41), we see that when we set p = p 0 and q = q 0 , where p 0 = √ α − i α, q 0 = √ α + i α, (44) the abov e equation would further simplify to  ∂ x 1 + i c∂ x − 1  m ( n,k ) i,j    p = p 0 , q = q 0 = 2 √ α i X µ =0 , µ : ev en 1 µ ! m ( n,k ) i − µ,j    p = p 0 , q = q 0 + 2 √ α j X l =0 , l : ev en 1 l ! m ( n,k ) i,j − l    p = p 0 , q = q 0 . (45) No w, w e restrict the general determinant (40) to τ n,k = det 1 ≤ i,j ≤ N  m ( n,k ) 2 i − 1 , 2 j − 1    p = p 0 , q = q 0  . (46) Then, using the contiguit y relation (45) as in Ref. [18], we get  ∂ x 1 + i c∂ x − 1  τ n,k = 4 √ α N τ n,k , whic h sho ws that the τ n,k function (46) satisfies the dimension reduction condition (42). When this dimension reduction equation is used to eliminate x − 1 from the higher-dimensional bilinear system (29)-(32), and in view of the parameter connection (41), w e get  D x 2 − D 2 x 1 + 2i αD x 1  τ n − 1 ,k +1 · τ n − 1 ,k = 0 , (47)  D x 2 − D 2 x 1  τ n,k · τ n − 1 ,k = 0 , (48) (i D x 1 − 1) τ n,k · τ n − 1 ,k + τ n − 1 ,k +1 τ n,k − 1 = 0 , (49) ( D 2 x 1 + i D x 1 + 2 α ) τ n,k · τ n − 1 ,k − (i D x 1 + 2 α ) τ n − 1 ,k +1 · τ n,k − 1 = 0 . (50) In addition, using Eq. (49), w e can replace the last bilinear equation (50) by D 2 x 1 τ n,k · τ n − 1 ,k − i D x 1 τ n − 1 ,k +1 · τ n,k − 1 + (2 α + 1)( τ n,k · τ n − 1 ,k − τ n − 1 ,k +1 · τ n,k − 1 ) = 0 . (51) In these reduced bilinear equations, the x − 1 deriv ative disapp ears. T o further reduce the bilinear system (47)-(49) and (51) to the original system (7)-(10), we set x 1 = x − 2 t, x 2 = i t. (52) 10 Under this v ariable relation, we hav e ∂ x 1 = ∂ x , ∂ x 2 = − i ∂ t − 2i ∂ x . (53) Inserting these equations into the bilinear system (47)-(49) and (51), and setting n = k = 0, we get  i D t + D 2 x + 2i(1 − α ) D x  g · ¯ f = 0 , (54)  i D t + D 2 x + 2i D x  f · ¯ f = 0 , (55) (i D x − 1) f · ¯ f + g ¯ g = 0 , (56) D 2 x f · ¯ f − i D x g · ¯ g + (2 α + 1)( | f | 2 − | g | 2 ) = 0 , (57) where f , ¯ f , g and ¯ g are defined as f = τ 0 , 0 , ¯ f = τ − 1 , 0 , g = τ − 1 , 1 , ¯ g = τ 0 , − 1 . (58) C. Complex conjugacy conditions Next, w e need to imp ose complex conjugacy conditions ¯ f = f ∗ and ¯ g = g ∗ , i.e., τ − 1 , 0 = τ ∗ 0 , 0 , τ 0 , − 1 = τ ∗ − 1 , 1 , (59) so that the bilinear system (54)-(57) would reduce to the original bilinear system (7)-(10). These conditions can b e satisfied by imp osing the parameter constrain t ξ 0 = η ∗ 0 . Indeed, under this constraint, since x 1 = x − 2 t is real, x 2 = i t , c = − i α are pure imaginary , and q 0 = p ∗ 0 , w e can easily show that h m ( n,k ) i,j i ∗    p = p 0 , q = q 0 = m ( − n − 1 , − k ) j,i    p = p 0 , q = q 0 . (60) Th us, τ ∗ n,k = τ − n − 1 , − k , i.e., the complex conjugacy conditions (59) hold. D. Rogue wa v e solutions in differential op erator form Finally , we need to introduce free parameters into rogue wa v es. Unlike all previous bilinear approaches [18, 25–30], w e will in tro duce free parameters through the arbitrary constant ξ 0 in Eq. (36). Sp ecifically , we choose ξ 0 as ξ 0 = ∞ X r =1 ˆ a r ln r  p − c p 0 − c  = ∞ X r =1 ˆ a r ln r  p + i α √ α  , (61) where ˆ a r are free complex constants. W e can show that rogue wa v es with this new parameterization can b e conv erted to those with the old parameterization through nontrivial parameter connections. But the new parameterization will drastically simplify rogue wa v e expressions. Putting all the ab o v e results together and setting x − 1 = 0, rational solutions to the GDNLS equations (2) are given b y the follo wing theorem. Theorem 3 The GDNLS e quations (2) admit r ational solutions u N ( x, t ) = e i(1 − γ − α ) x − i [ α 2 +2( γ − 2) α +1 − γ ] t ( f ∗ N ) γ − 1 g N f γ N , (62) wher e f N ( x, t ) = τ 0 , 0 , g N ( x, t ) = τ − 1 , 1 , (63) τ n,k = det 1 ≤ i,j ≤ N  m ( n,k ) 2 i − 1 , 2 j − 1  , (64) 11 the matrix elements in τ n,k ar e define d by m ( n,k ) i,j = [( p + i α ) ∂ p ] i i ! [( q − i α ) ∂ q ] j j ! " i p p + q  − p q  n  − p + i α q − i α  k e Θ( x,t ) #      p = p 0 , q = q 0 , (65) with Θ( x, t ) = ( p + q )( x − 2 t ) + ( p 2 − q 2 )i t + ∞ X r =1 ˆ a r ln r  p + i α √ α  + ∞ X r =1 ˆ a ∗ r ln r  q − i α √ α  , (66) p 0 , q 0 ar e given in Eq. (44), α > 0 , and ˆ a r ( r = 1 , 2 , . . . ) ar e fr e e c omplex c onstants. E. Explicit rogue w av e solutions through Sc hur polynomials The ab ov e rational solutions in Theorem 3 inv olve differential op erators, which make them less explicit. More seriously , such forms mak e analysis of those solutions difficult. F or instance, under such forms, it is difficult to prov e that they satisfy the b oundary conditions (5). In addition, it is difficult to determine the maximum p eak amplitudes for rogue wa ves of eac h order. Thus, in this subsection, we derive a more explicit form for these solutions, which is the one given in Theorem 1 earlier in the pap er. The technique we use is similar to that in Ref. [18]. The differential op erators in (65) can b e rewritten as (43), where ˆ p = p + i α and ˆ q = q − i α , and the m ( n,k ) term follo wing the differential op erators in (65) can b e rewritten as m ( n,k ) = i( ˆ p − i α ) ˆ p + ˆ q  − ˆ p − i α ˆ q + i α  n  − ˆ p ˆ q  k × exp ( ( ˆ p + ˆ q ) ( x − 2 t ) +  ˆ p 2 − ˆ q 2 − 2i α ( ˆ p + ˆ q )  i t + ∞ X r =1 ˆ a r ln r  ˆ p ˆ p 0  + ∞ X r =1 ˆ a ∗ r ln r  ˆ q ˆ q 0  ) , where ˆ p 0 = p 0 + i α and ˆ q 0 = q 0 − i α , i.e., ˆ p 0 = ˆ q 0 = √ α . Then, introducing the generator G of differential op erators ( ˆ p∂ ˆ p ) i ( ˆ q ∂ ˆ q ) j as G = ∞ X i =0 ∞ X j =0 ζ i i ! λ j j ! [ ˆ p∂ ˆ p ] i [ ˆ q ∂ ˆ q ] j , (67) and utilizing the formula [18] G F ( ˆ p, ˆ q ) = F  e ζ ˆ p, e λ ˆ q  , (68) w e get G m ( n,k )    ˆ p = ˆ p 0 , ˆ q = ˆ q 0 = e ζ / 2 (i e ζ / 2 + √ αe − ζ / 2 ) e ζ + e λ ( − 1) k e ( k + n 2 )( ζ − λ )  i e ζ / 2 + √ αe − ζ / 2 − i e λ/ 2 + √ αe − λ/ 2  n × exp ( √ α  e ζ + e λ  ( x − 2 t + 2 αt ) + α  e 2 ζ − e 2 λ  i t + ∞ X r =1 ( a r ζ r + a ∗ r λ r ) ) . Since m ( n,k )    ˆ p = ˆ p 0 , ˆ q = ˆ q 0 = ( − 1) k (i + √ α ) 2  i + √ α − i + √ α  n e 2 √ α ( x − 2 t +2 αt ) , w e ha ve 1 m ( n,k ) G m ( n,k )    ˆ p = ˆ p 0 , ˆ q = ˆ q 0 = 2 e ζ + e λ e ζ / 2+( k + n 2 )( ζ − λ )  i e ζ / 2 + √ αe − ζ / 2 i + √ α  n +1  − i + √ α − i e λ/ 2 + √ αe − λ/ 2  n × exp √ α  e ζ + e λ − 2  ( x − 2 t + 2 αt ) + α  e 2 ζ − e 2 λ  i t + ∞ X r =1 ( a r ζ r + a ∗ r λ r ) ! . (69) 12 No w, w e need to expand the right side of the ab ov e equation into pow er series of ζ and λ . F or this purp ose, we denote i e ζ / 2 + √ αe − ζ / 2 i + √ α = exp  ln  i e ζ / 2 + √ αe − ζ / 2 i + √ α  = exp ∞ X r =1 h r ζ r ! , where h r ( α ) is as defined in Eq. (15). The exp onen t in the most right-hand side of Eq. (69) can b e rewritten as exp ∞ X r =1 ζ r r !  √ α ( x − 2 t + 2 αt ) + 2 r i αt  + ∞ X r =1 λ r r !  √ α ( x − 2 t + 2 αt ) − 2 r i αt  + ∞ X r =1 ( a r ζ r + a ∗ r λ r ) ! , and the 2 / ( e ζ + e λ ) term can b e written as [18] 2 e ζ + e λ = ∞ X ν =0  ζ λ 4  ν exp ∞ X r =1 ( ν s r − c r ) ( ζ r + λ r ) − ζ 2 − λ 2 ! , where c r are T aylor co efficients of λ r in the expansion of ln cosh( λ/ 2), and s r are given in Eq. (15). Combining the ab o v e results, Eq. (69) b ecomes 1 m ( n,k ) G m ( n,k )    ˆ p = ˆ p 0 , ˆ q = ˆ q 0 = ∞ X ν =0  ζ λ 4  ν exp ∞ X r =1  x + r + ν s r  ζ r + ∞ X r =1  x − r + ν s r  λ r ! , (70) where x + r ( n, k ) and x − r ( n, k ) are defined as x + 1 ( n, k ) = √ α ( x − 2 t + 2 αt ) + 2i α t + ( n + 1) h 1 + k + n 2 − c 1 + ˆ a 1 , x − 1 ( n, k ) = √ α ( x − 2 t + 2 αt ) − 2i α t − nh ∗ 1 − k − 1 2 ( n + 1) − c 1 + ˆ a ∗ 1 , x + r ( n, k ) = 1 r !  √ α ( x − 2 t + 2 αt ) + 2 r i αt  + ( n + 1) h r − c r + ˆ a r , x − r ( n, k ) = 1 r !  √ α ( x − 2 t + 2 αt ) − 2 r i αt  − nh ∗ r − c r + ˆ a ∗ r . W e further define shifted parameters a 1 = ˆ a 1 − c 1 + 1 2 h 1 − 1 4 , a r = ˆ a r − c r + 1 2 h r . Then the ab ov e x + r and x − r reduce to those in Theorem 1. T aking the co efficients of ζ i λ j on b oth sides of Eq. (70), w e get m ( n,k ) i,j m ( n,k )   p = p 0 ,q = q 0 = min( i,j ) X ν =0 1 4 ν S i − ν  x + + ν s  S j − ν  x − + ν s  , where m ( n,k ) i,j is the matrix element defined in Eq. (65) of Theorem 3. Finally , w e define σ n,k = τ n,k  m ( n,k )   p = p 0 ,q = q 0  N . Then the matrix element in σ n,k is as giv en in Theorem 1. Since the bilinear equations (7)-(10) are in v arian t when f and g are divided by an arbitrary complex constan t multiplying an exp onen tial of a linear and real function in x and t , σ n,k then is also a solution to the GDNLS equations (2). Regarding b oundary conditions of these rational solutions, using the Sch ur p olynomial expressions in Theorem 1 and the same technique as in Ref. [18], we can show that when x or t approac hes infinity , f N ( x, t ) and g N ( x, t ) hav e the same leading term, which is also real. Th us, the rational solution (13) satisfies the b oundary condition (5), and is th us a rogue wa v e. Theorem 1 is then pro ved. 13 F. The parity-time-symmetric rogue wa v e In this subsection, we derive the parit y-time-symmetric rogue w av e and prov e Theorem 2. When w e set all a r = 0 in Theorem 1, x + r and x − r satisfy the following relations b x ± r ( x, t ) = − x ∓ r ( x, t ) , r ≥ 1 , where w e ha ve defined b f ( x, t ) ≡ f ∗ ( − x, − t ) for any function f ( x, t ). Thus, b x ± ( n, k ) + ν s = y ∓ ( n, k ) + ν s + z ∓ ( n ) , where v ectors y ± and z ± are defined as y ± =  − x ± 1 , x ± 2 , − x ± 3 , x ± 4 , · · ·  , z ± =  0 , − 2 x ± 2 , 0 , − 2 x ± 4 , 0 , · · ·  . Notice that ∞ X j =0 S j  b x ∓ + ν s  λ j = ∞ X j =0 S j  y ± + ν s + z ±  λ j = exp   ∞ X j =1  y ± j + ν s j + z ± j  λ j   = exp   ∞ X j =1  y ± j + ν s j  λ j   exp   ∞ X j =1 z ± j λ j   = ∞ X j =0 S j ( y ± + ν s ) λ j ∞ X j =0 S j ( z ± ) λ j = ∞ X j =0 X µ 1 + µ 2 = j S µ 1 ( y ± + ν s ) S µ 2 ( z ± ) λ j . Since s 1 = s 3 = · · · = s odd = 0 in view of Eq. (17), b y comparing the co efficien t of λ j on the tw o sides of this equation and utilizing Lemmas 2 and 3 in Ref. [28], w e get the relation S j  b x ∓ + ν s  = ( − 1) j [ j / 2] X µ =0 S µ ( w ± ) S j − 2 µ ( x ± + ν s ) , (71) where w ± =  − 2 x ± 2 , − 2 x ± 4 , · · ·  . Recall from Theorem 1 that σ n,k = det 1 ≤ i,j ≤ N   min(2 i − 1 , 2 j − 1) X ν =0 1 2 ν S 2 i − 1 − ν ( x + ( n, k ) + ν s ) 1 2 ν S 2 j − 1 − ν ( x − ( n, k ) + ν s )   , and b σ n,k is equal to the right side of the ab ov e equation with x ± replaced b y b x ± . By rewriting these t w o determinan ts in to 3 N × 3 N determinan ts as in [18], utilizing relations (71) and p erforming simple row manipulations, w e can quic kly sho w that b σ n,k = σ n,k . Th us, the solution u N ( x, t ) in Theorem 1 with all a r b eing zero satisfies the parity-time symmetry b u N = u N , i.e., u ∗ N ( − x, − t ) = u N ( x, t ). Theorem 2 is then prov ed. It turns out that the conv erse is also true, i.e., if a rogue wa ve u N ( x, t ) in Theorem 1 is parity-time-symmetric, then a 1 = a 3 = · · · = a odd = 0 [there is no restriction on the a ev en v alues b ecause the solution is indep endent of them, see Eq. (22)]. Our pro of is based on calculating the deriv ativ es of the p olynomial σ n,k with resp ect to the real part ξ 2 r − 1 and imaginary part η 2 r − 1 of the parameter a 2 r − 1 . Using Eqs. (20)-(21), we can show that each of ∂ ξ 2 r − 1 σ n,k and i ∂ η 2 r − 1 σ n,k con tains p o wer terms of ( x, t ) which are not parit y-time-symmetric. Thus, if any a odd is non-zero, the solution u N ( x, t ) would not b e parity-time-symmetric. VI. R OGUE W A VES THR OUGH A DIFFERENT KP-REDUCTION PROCEDURE As w e hav e mentioned in Sec. II, the GDNLS equations (2) admit tw o different bilinearizations. The first bilinear system is Eqs. (7)-(10), while the second bilinear system is Eqs. (8)-(10) and (11), i.e.,  i D t + D 2 x − 2i αD x  g · f = 0 , (72)  i D t + D 2 x + 2i D x  f · f ∗ = 0 , (73) (i D x − 1) f · f ∗ + | g | 2 = 0 (74) D 2 x f · f ∗ − i D x g · g ∗ + (2 α + 1)( | f | 2 − | g | 2 ) = 0 . (75) 14 Rogue w a v es in the GDNLS equations (2), as giv en in Theorem 1, can also b e deriv ed from this second bilinear system, but the corresp onding KP-reduction pro cedure is different. This will b e sho wn b elo w. This situation is analogous to m ulti-soliton solutions in the Sasa-Satsuma equation, which also admit tw o different bilinearizations and tw o different reduction procedures [55] A. Algebraic solutions for a higher-dimensional bilinear system First, w e consider the following higher-dimensional bilinear equations in the extended KP hierarc h y  D x 2 − D 2 x 1 − 2 dD x 1  τ n,k,l +1 · τ n,k,l = 0 , (76)  D x 2 − D 2 x 1  τ n,k,l · τ n − 1 ,k,l = 0 , (77)  cD x − 1 + 1  τ n − 1 ,k,l · τ n,k,l = τ n,k − 1 ,l τ n − 1 ,k +1 ,l , (78) ( cD x 1 D x − 1 − D x 1 − 2 c ) τ n,k,l · τ n − 1 ,k,l = ( D x 1 − 2 c ) τ n,k − 1 ,l · τ n − 1 ,k +1 ,l , (79) where c and d are arbitrary complex constants. The main difference b et ween these bilinear equations and the previous ones (29)-(32) is the introduction of the third index l in the τ function, whic h is necessary in order to reduce the first bilinear equation (76) to (72). Indeed, the previous tw o-index τ function (28) is unable to satisfy a higher-dimensional bilinear equation which can b e reduced to (72). W e can show that if functions m ( n,k,l ) i,j , ϕ ( n,k,l ) i and ψ ( n,k,l ) j of v ariables ( x − 1 , x 1 , x 2 ) satisfy the following differen tial and difference relations ∂ x 1 m ( n,k,l ) i,j = ϕ ( n,k,l ) i ψ ( n,k,l ) j , ∂ x 1 ϕ ( n,k,l ) i = ϕ ( n +1 ,k,l ) i , ∂ x 1 ψ ( n,k,l ) j = − ψ ( n − 1 ,k,l ) j , ∂ x 1 ϕ ( n,k,l ) i = cϕ ( n,k,l ) i + ϕ ( n,k +1 ,l ) i , ∂ x 1 ψ ( n,k,l ) j = − cψ ( n,k,l ) j − ψ ( n,k − 1 ,l ) j , ∂ x 1 ϕ ( n,k,l ) i = dϕ ( n,k,l ) i + ϕ ( n,k,l +1) i , ∂ x 1 ψ ( n,k,l ) j = − dψ ( n,k,l ) i − ψ ( n,k,l − 1) j , ∂ x 2 ϕ ( n,k,l ) i = ∂ 2 x 1 ϕ ( n,k,l ) i , ∂ x 2 ψ ( n,k,l ) j = − ∂ 2 x 1 ψ ( n,k,l ) j , ∂ x − 1 ϕ ( n,k,l ) i = ϕ ( n,k − 1 ,l ) i , ∂ x − 1 ψ ( n,k,l ) j = − ψ ( n,k +1 ,l ) j , (80) then the determinant τ n,k,l = det 1 ≤ i,j ≤ N  m ( n,k,l ) i,j  (81) w ould satisfy the new higher-dimensional bilinear system (76)-(79). No w, w e introduce the function m ( n,k,l ) as m ( n,k,l ) = i p p + q  − p q  n  − p − c q + c  k  − p − d q + d  l e ξ + η , where ξ = 1 p − c x − 1 + px 1 + p 2 x 2 + ξ 0 , η = 1 q + c x − 1 + q x 1 − q 2 x 2 + η 0 , and ξ 0 and η 0 are arbitrary complex constants. Then, b y defining m ( n,k,l ) i,j = A i B j m ( n,k,l ) , (82) where A i and B j are differential op erators as defined in Eq. (39), then this m ( n,k,l ) i,j , together with appropriately c hosen ϕ ( n,k,l ) i and ψ ( n,k,l ) j , satisfies those differential-difference equations (80), and thus the determinan t (81) satisfies the bilinear system (76)-(79) for arbitrary sequences of indices ( i 1 , i 2 , · · · , i N ; j 1 , j 2 , · · · , j N ). T o reduce the higher-dimensional bilinear system (76)-(79) to (72)-(75), we will set c = − i α, d = − i(1 + α ) . (83) 15 B. Dimension reduction Our dimension reduction is the same as b efore, i.e.,  ∂ x 1 + i c∂ x − 1  τ n,k,l = C τ n,k,l , (84) where C is a certain constant. The same calculations as in Sec. V B show that the determinan t τ n,k,l = det 1 ≤ i,j ≤ N  m ( n,k,l ) 2 i − 1 , 2 j − 1    p = p 0 , q = q 0  , (85) with p 0 , q 0 giv en b y Eq. (44), would satisfy this dimension reduction condition. Under this reduction, the bilinear equation (78) b ecomes (i D x 1 − 1) τ n,k,l · τ n − 1 ,k,l + τ n,k − 1 ,l τ n − 1 ,k +1 ,l = 0 , (86) and (79), combined with (86), reduces to D 2 x 1 τ n,k,l · τ n − 1 ,k,l + i D x 1 τ n,k − 1 ,l · τ n − 1 ,k +1 ,l = (2i c + 1)( τ n,k − 1 ,l · τ n − 1 ,k +1 ,l − τ n,k,l · τ n − 1 ,k,l ) . (87) C. The index reduction The key step to reduce the bilinear equation (76) to (72) is the observ ation that the curren t three-index τ function (85) admits the following index relation, τ n,k − 1 ,l = K N τ n − 1 ,k,l − 1 , K =  √ α + i √ α − i  2 . (88) Its pro of resem bles that in Ref. [27] for showing a similar index relation but for a different in tegrable equation. F rom the definition of m ( n,k,l ) i,j in Eq. (82), we hav e m ( n,k − 1 ,l ) i,j = A i B j m ( n,k − 1 ,l ) = A i B j  p q   − q + c p − c   p − d q + d  m ( n − 1 ,k,l − 1) . Defining H ( ˆ p ) = p ( p − d ) p − c , e H ( ˆ q ) = − q + c q ( q + d ) , where ˆ p = p − c and ˆ q = q + c , then m ( n,k − 1 ,l ) i,j = A i B j H ( ˆ p ) e H ( ˆ q ) m ( n − 1 ,k,l − 1) . F rom the Leibniz rule, we can rewrite the abov e equation as m ( n,k − 1 ,l ) i,j = i X ν =0 j X r =0 1 ν ! 1 r ! H ν ( ˆ p ) e H r ( ˆ q ) m ( n − 1 ,k,l − 1) i − ν,j − r , where functions H ν ( ˆ p ) and e H r ( ˆ q ) are defined as H ν ( ˆ p ) = ( ˆ p∂ ˆ p ) ν H ( ˆ p ) , e H r ( ˆ q ) = ( ˆ q ∂ ˆ q ) r e H ( ˆ q ) . In tro ducing t wo generators G 1 = ∞ X ν =0 ζ ν ν ! ( ˆ p∂ ˆ p ) ν , G 2 = ∞ X r =0 λ r r ! ( ˆ q ∂ ˆ q ) r , 16 and using the formula (68), w e get G 1 H ( ˆ p ) = H ( e ζ ˆ p ) = e ζ ˆ p + c ( c − d ) ˆ p e − ζ + 2 c − d, G 2 e H ( ˆ q ) = e H ( e λ ˆ q ) = − 1 e λ ˆ q + c ( c − d ) ˆ q e − λ − 2 c + d . F or the c hosen c, d v alues (83) and v alues ˆ p 0 = ˆ q 0 = √ α from (44), w e see that G 1 H ( ˆ p 0 ) and G 2 e H ( ˆ q 0 ) are even functions of ζ and λ , resp ectively . Th us, H 2 ν − 1 ( ˆ p 0 ) = e H 2 ν − 1 ( ˆ q 0 ) = 0 for all ν ≥ 1. Utilizing these results, we get the relation m ( n,k − 1 ,l ) i,j | p = p 0 ,q = q 0 = i X ν =0 , ν : ev en j X r =0 , r : ev en 1 ν ! 1 r ! H ν ( ˆ p ) e H r ( ˆ q ) m ( n − 1 ,k,l − 1) i − ν,j − r | p = p 0 ,q = q 0 . Th us,  m ( n,k − 1 ,l ) 2 i − 1 , 2 j − 1    p = p 0 , q = q 0  1 ≤ i,j ≤ N = L  m ( n − 1 ,k,l − 1) 2 i − 1 , 2 j − 1    p = p 0 , q = q 0  1 ≤ i,j ≤ N U, where L is a certain low er triangular matrix with H 0 ( ˆ p 0 ) on the diagonal, and U is a certain upp er triangular matrix with e H 0 ( ˆ q 0 ) on the diagonal. T aking determinants to this equation, w e get τ n,k − 1 ,l = h H 0 ( ˆ p 0 ) e H 0 ( ˆ q 0 ) i N τ n − 1 ,k,l − 1 , whic h is the same as (88) since H 0 ( ˆ p 0 ) e H 0 ( ˆ q 0 ) = K . D. Rogue wa v e solutions No w, we set x 1 = x − 2 t , x 2 = i t , c, d as in (83), and n = k = l = 0 in the ab ov e bilinear equations (76), (77), (86) and (87). Since τ 0 , 0 , 1 = ( K ) N τ − 1 , 1 , 0 due to the index relation (88), we find that when w e define f = τ 0 , 0 , 0 , ¯ f = τ − 1 , 0 , 0 , g = τ − 1 , 1 , 0 , ¯ g = τ 0 , − 1 , 0 , the abov e bilinear equations would b ecome  i D t + D 2 x − 2i αD x  g · f = 0 ,  i D t + D 2 x + 2i D x  f · ¯ f = 0 , (i D x − 1) f · ¯ f + g ¯ g = 0 , D 2 x f · ¯ f − i D x g · ¯ g + (2 α + 1)( f ¯ f − g ¯ g ) = 0 . (89) Notice that these ( f , ¯ f , g , ¯ g ) functions all hav e index l = 0. Thus, these functions are exactly the same as those giv en in Eq. (58) of the earlier section. Then, following the same complex-conjugacy reductions ¯ f = f ∗ and ¯ g = g ∗ as b efore, the bilinear system (89) reduces to Eqs. (72)-(75), and its rogue wa v e solutions are exactly as given in Theorems 1 and 3. VI I. CONCLUSIONS AND DISCUSSIONS In this article, w e ha ve derived general rogue wa v es in the GDNLS equations (1) by an improv ed bilinear KP reduction metho d. Since these GDNLS equations arise in m ultiple ph ysical situations and contain the Kaup-Newell equation, the Chen-Lee-Liu equation and others as sp ecial cases, these results would b e useful for rogue-wa v e gen- eration in suc h ph ysical systems. A main b enefit of this bilinear framework is that, rogue wa ves to all members of these GDNLS equations can b e expressed by the same bilinear solution. Compared to previous bilinear KP reduction metho ds for rogue w av es in other in tegrable equations, an imp ortan t improv ement in our current KP reduction tech- nique is a new parameterization of internal parameters in rogue wa ves. Under this new parameterization, the bilinear 17 solution is m uch simpler than b efore. In addition, the rogue w av e with the highest p eak amplitude at each order can b e easily obtained by setting all these in ternal parameters to zero. This wa y , the maximum p eak amplitude at order N is found to b e 2 N + 1 times the background amplitude, indep endent of the individual GDNLS equation and the bac kground wa v enum ber. W e hav e also found that these GDNLS equations can b e decomp osed into t wo different bilinear systems which require different KP reductions, but the resulting rogue wa v es are the same. Dynamics of rogue w a ves in the GDNLS equations is also analyzed. It is shown that the wa ven umber of the constan t background strongly affects the orientation and duration of the rogue wa ve. In addition, some new rogue patterns are presen ted. The GDNLS equations (1) considered in this article hav e the parameter requiremen t of a 6 = b , in which case these equations are gauge-equiv alen t to the deriv ative NLS equation of Kaup-Newell type (4) (see Sec. I I). If a = b , Eq. (1) is called the Kundu-Eckhaus equation in the literature [38]. The Kundu-Ec khaus equation is gauge-equiv alent to the NLS equation rather than the deriv ative NLS equation, and th us its rogue wa v es would b e different from those for the GDNLS equations (1) with a 6 = b . Rogue w av es in the Kundu-Eckhaus equation hav e b een studied by Darb oux transformation in [56–58]. In the bilinear framew ork, w e can deriv e general rogue wa v es in the Kundu-Eckhaus equation in a similar wa y as we did for the GDNLS equations (1) with a 6 = b . This deriv ation will be sketc hed in the app endix. Ac knowledgemen t The w ork of B.Y. and J.Y. is supp orted in part by the National Science F oundation (DMS-1910282) and the Air F orce Office of Scientific Research (F A9550-18-1-0098), and the work of J.C. is supp orted b y the National Natural Science F oundation of China (No.11705077). J.C. thanks J.Y. and the Universit y of V ermont for hospitality during his visit, where this work w as done. App endix: Bilinear deriv ation of rogue wa v es in the Kundu-Eckhaus equation When a = b , Eq. (1) b ecomes the Kundu-Eckhaus equation [38] i φ t + φ ξξ + ρ | φ | 2 φ + i a ( | φ | 2 ) ξ φ + 1 4 a 2 | φ | 4 φ = 0 . (90) Under a gauge transformation φ ( ξ , t ) = w ( ξ , t ) e − a 2 i R | w ( ξ ,t ) | 2 dξ , this Kundu-Ec khaus equation reduces to the NLS equation i w t + w ξξ + ρ | w | 2 w = 0 , (91) whose rogue w av es hav e b een derived before [12 – 18]. T o directly obtain rogue w av es in the Kundu-Ec khaus equation (90) without the use of the ab ov e gauge transformation, w e can apply a similar bilinear approach as we did for the a 6 = b case in the main text of this article. Sp ecifically , through a scaling of ( φ, ξ , t, a ) together with a Galilean transformation, w e can normalize ρ = 2 in Eq. (90), and the boundary conditions of its rogue wa v es can be normalized as φ ( ξ , t ) → e i ( 2 t − 1 2 aξ ) , ( ξ , t ) → ∞ . (92) Then, w e emplo y a bilinear v ariable transformation φ ( ξ , t ) = e i [ 2 t − 1 2 a [ ξ +(ln f ) ξ ] ] g f , (93) where f is a real function, and g a complex function. Under this transformation, the Kundu-Eckhaus equation (90) can be split into the follo wing three bilinear equations,  i D t + D 2 ξ  g · f = 0 , (94) ( D 2 ξ + 2) f · f = 2 | g | 2 , (95) D ξ D t f · f = 2i D ξ g · g ∗ . (96) One can recognize that the first tw o bilinear equations are the ones for the NLS equation (91) with ρ = 2 [18]. It turns out that the ( f , g ) solutions for rogue wa ves of the NLS equation also satisfy the third bilinear equation abov e, 18 and thus rogue wa v es for the Kundu-Eckhaus equation (90) are given by (93), where ( f , g ) are those for the NLS equation (91). The reason for this is that, under the same differential and difference relations of τ functions listed in Eq. (3.7) of Ref. 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