Research Article | Open Access
Two Parameters Deformations of Ninth Peregrine Breather Solution of the NLS Equation and Multi-Rogue Waves
This paper is a continuation of a recent paper on the solutions of the focusing NLS equation. The representation in terms of a quotient of two determinants gives a very efficient method of determination of famous Peregrine breathers and their deformations. Here we construct Peregrine breathers of order and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.
From the fundamental work of Zakharov and Shabat in 1972 who solved the nonlinear Schrödinger equation (NLS) using the inverse scattering method, a lot of studies have been carried out on this equation. Its and Kotlyarov studied the case of periodic and almost periodic algebrogeometric solutions to the focusing NLS equation and constructed these solutions in 1976 . Peregrine constructed the first quasi-rational solutions of NLS equation in 1983, nowadays called worldwide Peregrine breathers. In 1985, Akhmediev et al. obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather . Other families of higher order were constructed in a series of articles by Akhmediev et al. [3, 4] using Darboux transformations.
In 2010, it has been shown in  that rational solutions of the NLS equation can be written as a quotient of two Wronskians. Recently, in  a new representation of the solutions of the NLS equation has been constructed in terms of a ratio of two Wronskian determinants of even order composed of elementary functions; the related solutions of NLS are of order . When we perform the passage to the limit when some parameter tends to , we got families of multi-rogue wave solutions of the focusing NLS equation depending on a certain number of parameters. It allows to recognize the famous Peregrine breather  and also higher order Peregrine’s breathers constructed by Akhmediev et al. [3, 8].
Recently, another representation of the solutions of the focusing NLS equation, as a ratio of two determinants, has been given in  using generalized Darboux transform.
A new approach has been done in , which gives a determinant representation of solutions of the focusing NLS equation, obtained from Hirota bilinear method, derived by reduction of the Gram determinant representation for Davey-Stewartson system.
Here, we construct the breather of order , which shows the efficiency of this method.
2. Expression of Solutions of NLS Equation in terms of Wronskian Determinant and Quasi-Rational Limit
2.1. Solutions of the NLS Equation in terms of Functions
The solution of the NLS equation is given in terms of truncated theta function by (see ) where In this formula, , , , and are functions of the parameters , ; they are defined by the formulas The parameters , , are real numbers such that Condition (5) implies that Complex numbers are defined in the following way: , , are arbitrary real numbers.
2.2. Relation between and Fredholm Determinant
Let be the unit matrix and the matrix defined by Then has the following form: From the beginning of this section, has the same expression as in (10), so we have clearly the equality Then the solution of NLS equation takes the form
2.3. Link between Fredholm Determinants and Wronskians
We consider the following functions: We use the following notations: is the wronskian .
We consider matrix defined by Then we have the following statement.
Theorem 1. Consider where
Proof. We start to remove factor in each row in the wronskian for .
Then withThe determinant can be written as where , , and , , , , , and , , .
Denoting , , the determinant of is clearly equal to
Then we use the following lemma.
Lemma 2. Let , , and , the matrix formed by replacing the th row of by the th row of . Then
Proof. For , the transposed matrix in the cofactors of , we have the well-known formula .
So it is clear that .
The general term of the product can be written as We get Thus, .
According to the relation (22) of the previous lemma, we get where is the matrix formed by replacing the th row of by the th row of defined previously.
We compute and we get We can simplify the quotient So can be expressed as Then dividing each column by , , and multiplying each row by , , we get and therefore the wronskian can be written as It follows that So, the solution of NLS equation takes the form
2.4. Wronskian Representation of Solutions of NLS Equation
From the previous section, we get the following result.
Theorem 3. Function defined by is a smooth solution of the focusing NLS equation depending on two real parameters, and .
2.5. Quasi-Rational Solutions of NLS Equation in terms of a Limit of a Ratio of Wronskian Determinants
In the following, we take the limit when the parameters for and for .
For simplicity, we denote the term by .
We consider the parameter written in the form When goes to , we realize limited expansions at order , for , of the terms The parameters and , for , are chosen in the form Then we have the following result.
3. Quasi-Rational Solutions of Order 9
Wa have already constructed in  solutions for the cases until , and this method gives the same results. We do not reproduce it here. We only give solutions of NLS equation in the case .
Because of the length of the expressions of polynomials and in the solutions of the NLS equation defined by we only give them in the appendix. In the following cases, we only give the plots for the modulus of in the coordinates.
For , , we obtain Akhmediev’s breather; we get the expected amplitude of for the spike (Figure 1).
If we choose , , we obtain Figure 2.
If we choose , , we have Figure 3.
It can be noted that Figures 2 and 3 are closely analogous with Figure 2(b) in paper  of Kedziora et al. In that work (), it was pointed out that the shift (here corresponding to and nonzero) pulls out a ring of fundamental rogue elements, corresponding to 15 of them there and to 17 here. It leaves behind a rogue wave of order , that is, 6 there (amplitude = 13) and 7 here (amplitude = 15). Of course, Figure 1 here is analogous with Figure 2(a) there (amplitudes 19 and 17, resp.).
The method used in the present paper provides a powerful tool as the explicit analytical formulation of the ninth order shows it. To my knowledge, it is the first time that the analytical expression of the Peregrine breather of order nine is presented.
It confirms the conjecture about the shape of the breather in the coordinates, the maximum of amplitude equal to , and the degree of polynomials in and here equal to . For and nonzero, the maximum is less than that, as discussed above and seen in Figures 2 and 3.
In the following, we choose all the parameters and equal to ; here .
The solution of NLS equation takes the form with