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We study the Peregrine rogue waves within the framework of Derivative Nonlinear Schrödinger equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond pulses in nonlinear optics. The interaction and degeneration of two soliton-like solutions and its relations for the breather solution have been analyzed. The Peregrine rogue waves have been considered from the two kinds of formation processes: it can be generated through the limitation of the infinitely large period of the breather solutions, and it can be interpreted as the soliton-like solutions with different polarities. As a special example, a special Peregrine rogue wave is generated by a breather solution and phase solution, which is given by the trivial seed (zero solution).

In the past years, rogue waves, commonly defined as gigantic waves appearing from nowhere and disappearing without trace, have attracted a lot of attention in deep ocean waves [

i ψ t + 1 2 ψ x x + ψ 2 ψ ∗ = 0 , (1)

is commonly considered as a classic model to describe rogue waves. As it has a rich of many exact solutions due to its integrability, the Peregrine soliton [

ψ = ( 1 − 4 1 + 2 i t 1 + 4 t 2 + 4 x 2 ) e i t , (2)

possesses a high amplitude and two hollows and is the prototype of the rogue waves, which “appear from nowhere and disappear without a trace” [

Considering the generalization of the type of NLS equation, the study of rogue waves in the Derivative Nonlinear Schrödinger(DNLS) equation has also caused a lot of research [

i q t − q x x + i ( q 2 q ∗ ) x = 0, (3)

is originated from nonlinear optics and plasma physics. Here “*” denotes the complex conjugation, and subscript of x (or t) denotes the partial derivative with respect to x (or t). In nonlinear optics, the DNLS equation is used to describe the propagation of sub-picosecond or femtosecond pulses in optical fibers [

The structure of this paper is as follows. In Section 2, we give the analytical form of the Peregrine rogue waves by DT from a periodic solution of the DNLS equation. Based on the explicit expression and their formation process, we can get the relations between breather solutions, phase solutions, soliton solutions and Peregrine rogue waves. In Section 3, The interaction and the degeneration mechanism of two soliton-like solutions and their key properties such as its relations for the breather solution are discussed. In the limitation λ 1 → λ c 1 and λ 2 → λ c 2 ( λ c 1 = 1 2 ( c 2 − 2 a − c ) i , λ c 2 = 1 2 ( − c 2 − 2 a − c ) i ), the two soliton-like solutions gradually degenerate into the Peregrine rogue waves under the condition 3 8 c 2 < a ≤ 1 2 c 2 . Finally, we summarize our main results in Section 4.

The analytical form of Peregrine rogue waves q r of the DNLS equation is

q r = c ( − R 1 − 4 + 8 i c 2 a t + i R 2 ) ( R 1 − i R 2 ) ( R 1 + i R 2 ) 2 exp ( i a ( a t + x − c 2 t ) ) , R 1 = − 8 c 2 a 3 t 2 + 12 c 4 a 2 t 2 − 8 c 2 a 2 t x − 6 c 6 a t 2 − 2 c 2 a x 2 + 8 c 4 a t x − 1, R 2 = 4 c 2 a t + 2 c 2 x − 6 c 4 t . (4)

which includes both quasi-rational bright-dark solitons and Peregrine rogue waves [

amplitude of | q r | 2 occurs at ( x = − 3 ( − c 2 + 2 a ) a c 3 32 a − 12 c 2 , t = 1 a c 3 32 a − 12 c 2 ) and ( x = 3 ( − c 2 + 2 a ) a c 3 32 a − 12 c 2 , t = − 1 a c 3 32 a − 12 c 2 ), and is equal to 0. Obviously, this quasi-rational solution is a Peregrine rogue wave. In

formed by two kinds of excitation mechanism. The specific formation mechanism is described in more detail by the collisions of two soliton-like structures in the next part.

The DNLS equation [

∂ x ψ = ( J λ 2 + Q λ ) ψ = U ψ , (5)

∂ t ψ = ( 2 J λ 4 + V 3 λ 3 + V 2 λ 2 + V 1 λ ) ψ = V ψ , (6)

with

ψ = ( ϕ φ ) , J = ( i 0 0 − i ) , Q = ( 0 q r 0 ) ,

V 3 = 2 Q , V 2 = J q r , V 1 = ( 0 − i q x + q 2 r i r x + r 2 q 0 ) .

Here λ , an arbitrary complex number, is called the eigenvalue (or the spectral parameter), and ψ is called the eigenfunction associated with the eigenvalue λ of the KN Lax pair.

Next we give the general forms of the N-order soliton solutions [

q [ n ] = Ω 11 2 Ω 21 2 q + 2 i Ω 11 Ω 12 Ω 21 2 . (7)

Here, 1) for n = 2 k ,

Ω 11 = | λ 1 n − 1 φ 1 λ 1 n − 2 ϕ 1 λ 1 n − 3 φ 1 ⋯ λ 1 φ 1 ϕ 1 λ 2 n − 1 φ 2 λ 2 n − 2 ϕ 2 λ 2 n − 3 φ 2 ⋯ λ 2 φ 2 ϕ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n − 1 φ n λ n n − 2 ϕ n λ n n − 3 φ n ⋯ λ n φ n ϕ n | , (8)

Ω 12 = | λ 1 n ϕ 1 λ 1 n − 2 ϕ 1 λ 1 n − 3 φ 1 ⋯ λ 1 φ 1 ϕ 1 λ 2 n ϕ 2 λ 2 n − 2 ϕ 2 λ 2 n − 3 φ 2 ⋯ λ 2 φ 2 ϕ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n ϕ n λ n n − 2 ϕ n λ n n − 3 φ n ⋯ λ n φ n ϕ n | ,

Ω 21 = | λ 1 n − 1 ϕ 1 λ 1 n − 2 φ 1 λ 1 n − 3 ϕ 1 ⋯ λ 1 ϕ 1 φ 1 λ 2 n − 1 ϕ 2 λ 2 n − 2 φ 2 λ 2 n − 3 ϕ 2 ⋯ λ 2 ϕ 2 φ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n − 1 ϕ n λ n n − 2 φ n λ n n − 3 ϕ n ⋯ λ n ϕ n φ n | ,

2) for n = 2 k + 1 ,

Ω 11 = | λ 1 n − 1 φ 1 λ 1 n − 2 ϕ 1 λ 1 n − 3 φ 1 ⋯ λ 1 ϕ 1 φ 1 λ 2 n − 1 φ 2 λ 2 n − 2 ϕ 2 λ 2 n − 3 φ 2 ⋯ λ 2 ϕ 2 φ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n − 1 φ n λ n n − 2 ϕ n λ n n − 3 φ n ⋯ λ n ϕ n φ n | , (9)

Ω 12 = | λ 1 n ϕ 1 λ 1 n − 2 ϕ 1 λ 1 n − 3 φ 1 ⋯ λ 1 ϕ 1 φ 1 λ 2 n ϕ 2 λ 2 n − 2 ϕ 2 λ 2 n − 3 φ 2 ⋯ λ 2 ϕ 2 φ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n ϕ n λ n n − 2 ϕ n λ n n − 3 φ n ⋯ λ n ϕ n φ n | ,

Ω 21 = | λ 1 n − 1 ϕ 1 λ 1 n − 2 φ 1 λ 1 n − 3 ϕ 1 ⋯ λ 1 φ 1 ϕ 1 λ 2 n − 1 ϕ 2 λ 2 n − 2 φ 2 λ 2 n − 3 ϕ 2 ⋯ λ 2 φ 2 ϕ 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ λ n n − 1 ϕ n λ n n − 2 φ n λ n n − 3 ϕ n ⋯ λ n φ n ϕ n | ,

Note that the eigenfunction ψ k = ( ϕ k φ k ) associated with the eigenvalue λ k has the following properties [

1) ϕ k ∗ = φ k , λ k = − λ k ∗ ;

2) ϕ k ∗ = φ l , φ k ∗ = ϕ l , λ k ∗ = − λ l , where k ≠ l .

Based on the N-order solutions of the DNLS equation by determinant expression, we can get

q [ 2 ] = ( λ 1 φ 1 ϕ 2 − λ 2 ϕ 1 φ 2 ) 2 ( − λ 2 ϕ 2 φ 1 + λ 1 φ 2 ϕ 1 ) 2 q + 2 i ( λ 1 2 − λ 2 2 ) ϕ 1 ϕ 2 ( λ 1 φ 1 ϕ 2 − λ 2 ϕ 1 φ 2 ) ( − λ 2 ϕ 2 φ 1 + λ 1 φ 2 ϕ 1 ) 2 , (10)

with ϕ 1 and φ 1 given by Equation (11).

Set a and c to be two real constants, substituting q = c exp ( i ( a x + ( − c 2 + a ) a t ) ) into the spectral problem Equation (5) and Equation (6), the eigenfunction ψ k [

( ϕ k ( x , t , λ k ) φ k ( x , t , λ k ) ) = ( ϖ 1 ( x , t , λ k ) [ 1, k ] + ϖ 2 ( x , t , λ k ) [ 1, k ] + ϖ 1 ∗ ( x , t , − λ k ∗ ) [ 2, k ] + ϖ 2 ∗ ( x , t , − λ k ∗ ) [ 2, k ] ϖ 1 ( x , t , λ k ) [ 2, k ] + ϖ 2 ( x , t , λ k ) [ 2, k ] + ϖ 1 ∗ ( x , t , − λ k ∗ ) [ 1, k ] + ϖ 2 ∗ ( x , t , − λ k ∗ ) [ 1, k ] ) . (11)

Here

( ϖ 1 ( x , t , λ k ) [ 1 , k ] ϖ 1 ( x , t , λ k ) [ 2 , k ] ) = ( exp ( s ( x + 2 λ k 2 t + ( − c 2 + a ) t ) 2 + 1 2 ( i ( a x + ( − c 2 + a ) a t ) ) ) i a − 2 i λ k 2 + s 2 λ k c exp ( s ( x + 2 λ k 2 t + ( − c 2 + a ) t ) 2 − 1 2 ( i ( a x + ( − c 2 + a ) a t ) ) ) ) ,

( ϖ 2 ( x , t , λ k ) [ 1, k ] ϖ 2 ( x , t , λ k ) [ 2, k ] ) = ( exp ( − s ( x + 2 λ k 2 t + ( − c 2 + a ) t ) 2 + 1 2 ( i ( a x + ( − c 2 + a ) a t ) ) ) i a − 2 i λ k 2 − s 2 λ k c exp ( − s ( x + 2 λ k 2 t + ( − c 2 + a ) t ) 2 − 1 2 ( i ( a x + ( − c 2 + a ) a t ) ) ) ) ,

s = − a 2 − 4 λ k 4 − 4 λ k 2 ( c 2 − a ) .

1) Breather solution: Under the choice one paired eigenvalue λ 1 = α 1 + i β 1 and λ 2 = − α 1 + i β 1 and the eigenfunction ψ k associated with λ k from a periodic solution, then the breather solution has the following form as the Equation (53) from [

2) Two soliton-like solutions: Under the choice λ 1 = i β 1 , λ 2 = i β 2 and the eigenfunction ψ k associated with λ k from a periodic solution, then

| q s p | 2 = c 2 + 4 ( β 1 2 − β 2 2 ) 2 ( β 2 g 1 ∗ g 1 − β 1 g 2 ∗ g 2 ) ( β 2 g 1 g 1 ∗ − β 1 g 2 g 2 ∗ ) − R e ( 4 c ( β 1 2 − β 2 2 ) β 1 g 1 ∗ g 1 − β 2 g 2 ∗ g 2 ) , (12)

g i = 2 ( 1 + 2 β i 2 + a 2 c β i ) cosh ( 1 2 h i ( x − 2 β i 2 t + ( − c 2 + a ) t ) ) + i h i c β i sinh ( 1 2 h i ( x − 2 β i 2 t + ( − c 2 + a ) t ) ) ,

h i = 4 β i 2 c 2 − ( 2 β i 2 + a ) 2 , i = 1 , 2.

In the limitation β 1 → 1 2 ( c 2 − 2 a − c ) and β 2 → 1 2 ( − c 2 − 2 a − c ) , ( λ c 1 = 1 2 ( c 2 − 2 a − c ) i , λ c 2 = 1 2 ( − c 2 − 2 a − c ) i , the values of spectral parameters are the zero point of h i and give the soliton-like solutions with different polarities), the two soliton-like solutions | q s p | 2 gradually degenerate into the Peregrine rogue waves (see in

3) A special example: Based on the above two mechanisms, we found that 1 2 c 2 = a is boundary points. In order to better understand this state, we consider the equivalent solution in this case (when 1 2 c 2 = a , the seed solution has the following form: q = c exp ( i ( 1 2 c 2 x − 1 4 c 4 t ) ) , which can be given by (13) with n = 1 ).

q m s = 2 i Ω 11 Ω 12 Ω 11 ∗ 2 , (13)

Ω 11 = | λ 1 2 φ 1 λ 1 ϕ 1 φ 1 λ 2 2 φ 2 λ 2 ϕ 2 φ 2 λ 3 2 φ 3 λ 3 ϕ 3 φ 3 | ,

Ω 12 = | λ 1 3 ϕ 1 λ 1 ϕ 1 φ 1 λ 2 3 ϕ 2 λ 2 ϕ 2 φ 2 λ 3 3 ϕ 3 λ 3 ϕ 3 φ 3 | ,

ϕ 1 = exp [ i ( λ 1 2 x + 2 λ 1 4 t ) ] , φ 1 = exp [ − i ( λ 1 2 x + 2 λ 1 4 t ) ] , λ 1 = i h ,

ϕ 2 = exp [ i ( λ 2 2 x + 2 λ 2 4 t ) ] , φ 2 = exp [ − i ( λ 2 2 x + 2 λ 2 4 t ) ] , λ 2 = i l + s k ,

ϕ 3 = exp [ i ( λ 3 2 x + 2 λ 3 4 t ) ] , φ 3 = exp [ − i ( λ 3 2 x + 2 λ 3 4 t ) ] , λ 3 = i l − s k , s k ∈ R ∪ i R .

A special Peregrine rogue generated by a breather solution and phase solution (see

Based on the above analysis, we can get the relations between breather solutions, phase solutions, soliton solutions and rogue waves. The condition a > 3 8 c 2 on (c, a)-plane in

In the paper, we provide the formation mechanism of Peregrine rogue waves of the DNLS equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond pulses in nonlinear optics.

The bound state of two soliton-like solutions is figuratively illustrated in

This work is supported by the National Natural Science Foundation of China under Grant No. 11601187, Natural Science Foundation of Ningbo under Grant No. 2018A610197 and Major SRT Project of Jiaxing University.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhou, H.Q., Xu, S.W. and Li, M.H. (2020) Peregrine Rogue Waves Generated by the Interaction and Degeneration of Soliton-Like Solutions: Derivative Nonlinear Schrödinger Equation. Journal of Applied Mathematics and Physics, 8, 2824-2835. https://doi.org/10.4236/jamp.2020.812208