## Abstract

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution and *n*-fold DT *T*_{n} are given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues *k*=1,2,3,…, all these parameters will be defined latter. These solutions have a parameter *β*, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the *β* on the RW solution. We observe two interesting results on localization characters of *β*, such that if *β* is increasing from *a*/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if *β* is increasing to *a*/2. We define an area of the RW solution and find that this area associated with *c*=1 is invariant when *a* and *β* are changing.

## 1. Introduction

The possibility of propagation of soliton in optical fibre was introduced by Hasegawa & Tappert in 1973 [1,2] and was for the first time achieved in an experiment [3] 7 years later. At a very early stage of the research, an optical soliton is regarded as a result of the delicate balance between the dispersion and the cubic (Kerr) nonlinearity of the picosecond optical pulse propagation through monomode fibre without the inclusion of higher order nonlinear effects, which is described by the famous nonlinear Schrödinger (NLS) equation [4,5]
*q*=*q*(*x*,*t*) represents the envelope of an electric field, *t* denotes the normalized spatial variable and *x* is the normalized time variable. In optics, the squared modulus of the amplitude |*q*|^{2} usually denotes a measurable quantity, optical power (or intensity). The NLS equation is a well-known integrable system in mathematical physics possessing soliton and breather solutions, and also multi-Hamiltonian structures [6–8]. By now, the optical soliton is one of the promising information carriers in the futuristic optical fibre communication system because it can propagate to a long distance without distortion of profile and energy conservation [9–15]. This is a rare and direct example that an abstract mathematically discovered concept is converted into an technologically important invention, i.e. one kind of special highly localized solution called soliton of nonlinear partial differential equation, which can be used for high bit rate ultra-fast technology in optical fibre communication or for the generation of a supercontinuum white light source.

As the vast social demand for high bit rates in optical fibre communication and the advances in the area of laser technology, it is indeed necessary and possible to generate shorter (femtosecond, even attosecond) pulses with high frequency in fibre by increasing the intensity of the incident light power. This boosts greatly the study of higher order nonlinear effects in optics, which is modelled by the NLS with cubic-quintic terms in many cases. It was our prime aim to investigate the integrable case of cubic-quintic NLS with second-order dispersion and to consider the effects of the quintic term on the solutions. It is highly non-trivial to add quintic terms to the NLS equation without losing the integrability because of many combinations of the cubic and quintic terms and their derivatives. The simplest case of this consideration is an extended NLS formed by adding a quintic term |*u*|^{4}*u*, which is a special case of complex cubic-quintic Ginzburg–Landau equation and is not an integrable system. However, we indeed find one cubic-quintic NLS, i.e. the Kundu–Eckhaus (KE) equation
*A*_{A}(or *α*_{N}) involved with complicated integrations, which produces the difficulty in the construction of multi-fold DT. Therefore, the DT [20,25] of the KE equation is not fully understood and then needed to be improved by eliminating integration in order to get higher order RWs.

In recent years, the concept of a RW, which was first introduced to describe a suddenly appeared high-wall of water in deep ocean [26–29], has been gradually extended to different fields [30–32]. There are three main propelling facts of the active research of the RW recently: the first experimental observation [33] during 2007, extremely destructive high power in the ocean and possible application to generate ultra-high-energy pulse in the optical system. Although the first-order RW solution of the NLS was first reported by Peregrine in the year 1983 [34], experimental validations of such RW model have been successfully conducted in nonlinear fibres [35], in water wave tanks [36], and in plasmas [37] during 2010 and 2011, respectively. However, to date, there is no well-defined mathematical definition of the RW solution, although Akhmediev *et al.* [38] have provided a vivid description of the rogue characteristics—‘appear from nowhere and disappear without a trace’. According to their predictions, the featured properties are summarized by: (i) (quasi-) rational solution (or equivalently rational modulus of solution); (ii) localized doubly in both time and space; and (iii) large amplitude (the peak has a height at least three times of background) allocated with one hole in each side, which can be regarded as ‘definition’ of the first-order RW solution and is widely used by the scientific community working on nonlinear waves. Naturally, the higher order RW solutions have more peaks and exhibit several interesting patterns [39–44]. In addition to the NLS equation, there are many other equations admitting RW (or Peregrine-type) solutions such as the modified Korteweg–de Vries equation, the Fokas–Lenells equation, the derivative NLS equation, the long-wave–short-wave resonance equation, the vector NLS, the Davey–Stewartson equation and the KP-I equation [45–59], etc.

In this paper, we shall study the RW solutions of the KE equation motivated by the following problems.

— The construction of the DT for the KE equation was first realized in [20]. As we have mentioned above, it is involved with extremely complicated integrals to get the overall factor

*A*_{N}(see eqns (2.11) and (2.23) of [20]), and then needed to be improved by finding explicit expression of*A*_{N}. The improvement is highly non-trivial because two partial differential equations in terms of*A*_{N}are very complicated for multi-fold transformation and non-zero ‘seed’ solution. Moreover, this improvement is necessary to get explicit forms of solution by multi-fold DT from non-zero ‘seed’ solution.— Very recently, the first- and second-order RW solutions of the KE equation have been presented in [25] by using the DT introduced in [20]. Due to the limitation of representation of the DT discussed above, they have not constructed the higher order RW solutions of the KE equation. So getting the higher order RW solutions and their evolution in terms of different patterns are actually challenging uphill problems.

— We have defined two characters (the length and the width) [53] of the first-order RW solution for the NLS equation. Naturally, can we define an area of this doubly localized solution of the KE equation on the (

*t*,*x*) plane?— How the parameter

*β*in the KE equation affects the localization characters of the RW solution? Is it possible to answer this question by an analytical and a graphical way?

In this paper, we solve these four problems. The rest of the paper is organized as follows. In §2, we present the determinant expression of the *n*-fold DT and the formula of *n*th-order solutions of the KE equation. In §3, we prove the *n*-fold DT and the formula of the *n*th order solution in detail. In §4, we provide the first-order breather, first-order RW and fifth-order RWs. The phase difference between two first-order RWs of the NLS and the KE is discussed in detail. In §5, we define three localization characters, i.e. length, width and area of the first-order RW solution of the KE, and then study the impact of the *β* on them in an analytical and a graphical way. Finally, conclusions and discussion are given in §6.

## 2. The *n*-fold Darboux transformation for the Kundu–Eckhaus equation

In this section, we consider the *n*-fold DT for the KE equation, which is the compatibility condition of the following Lax pair [18]:
**over-bar** means complex conjugation, _{k}, which is used to generate the determinant representation of the *n*-fold DT below. For the eigenfunctions of Lax pair equations, there exists a following important property:

### Proposition 2.1

*Set* *be an eigenfunction associated with* λ, *then* *is an eigenfunction of* *Further, by linear combination*, *is a new eigenfunction of* λ. *Here, k*_{1} *and k*_{2} *are two constants*.

### Proof.

By a direct calculation, it is trivial to verify

According to proposition 2.1, we shall select eigenfunctions as next remark.

### Remark 2.2

Set _{2k−1}, *M*.

### Theorem 2.3

*The n-fold DT for KE equation is**with*
*for i*=0,1,…,*n*−1,
*and
*

Here, *A*_{2i+1} is the submatrix of *W*_{2i+2} by deleting the (2*i*+2)th column and the (2*i*+2)th row, *B*_{2i+1} is the submatrix of *W*_{2i+2} by deleting the (2*i*+1)th column and the (2*i*+1)th row, *C*_{2i+1} is the submatrix of *W*_{2i+2} by deleting the (2*i*+2)th column and the (2*i*+1)th row, *D*_{2i+1} is the submatrix of *W*_{2i+2} by deleting the (2*i*+1)th column and the (2*i*+2)th row. If *i*=0, we define |*W*_{0}|=1, such that

According to the representation of the *n*-fold DT in the above theorem, we get new solution *u*^{[n]} from ‘seed’ solutions *u*.

### Theorem 2.4

*The nth-order solution u*^{[n]} *generated by the n-fold DT is
**with
*

### Remark 2.5

The formula of *H*^{[i]} in equation (2.3) is very crucial to get the explicit form of the overall factor *u*^{[n]} from arbitrary initial ‘seed’ solutions *u* under the special selection in remark 2.2 as *k*=1,2,…,*n*.

## 3. Derivation of the *n*-fold DT

In this section, we derive the *n*-fold DT and the *n*th-order solutions for the KE equation given by theorem. In order to obtain the *n*-fold DT, we consider the one- and twofold DT at first, and then the *n*-fold DT can be generated by mathematical iteration.

### (a) The onefold Darboux transformation

We consider the simplest onefold *T*. Comparing to the DT of the NLS equation [43,60], we set that *T* has the following form:
*a*_{1},*b*_{1},*c*_{1},*d*_{1},*a*,*b*,*c*,*d* are functions of *x* and *t*. Moreover, there exists *Φ*^{[1]}=*T*_{1}*Φ* satisfying the following conditions: *M*^{[1]} and *N*^{[1]} have the same form as *M* and *N* except that *u* and *u*^{[1]} and

### Lemma 3.1

*The onefold DT for the KE equation can be defined as follows*:
*with H*^{[0]} *is the function of x and t*, *and the new solution u*^{[1]} *is defined by*
*under the selection*

### Proof.

Let matrix *Z*(λ)=(*Z*_{ij})=*T*_{1x}+*T*_{1}*M*−*M*^{[1]}*T*_{1}=0 (*i*,*j*=1,2) and substitute *T*_{1} into *Z*, then collecting the different coefficients of λ, we obtain that *b*_{1} and *c*_{1} are equal to zero,
*d*_{1x},*a*_{x},*b*_{x},*c*_{x},*d*_{x}, respectively. It is easy to know that *a*_{1}*d*_{1})_{x}=0. Based on the above results and taking the similar procedure as used above to get the second expression in equation (3.2), we get
*a*_{1}*d*_{1})_{t}=0, and other equations. Without loss of generality, we can set *a*_{1}=*H*^{[0]}, *d*_{1}=1/*H*^{[0]}, *a*=*a*_{0}*H*^{[0]}, *b*=*b*_{0}*H*^{[0]}, *c*=*c*_{0}/*H*^{[0]}, *d*=*d*_{0}/*H*^{[0]}, here *H*^{[0]}, *a*_{0}, *b*_{0}, *c*_{0}, *d*_{0} are functions of *x* and *t*. In particular, *T*_{1}(λ;λ_{1},λ_{2})*Φ*_{1}|_{λ=λ1}=0, *T*_{1}(λ;λ_{1},λ_{2})*Φ*_{2}|_{λ=λ2}=0, and solving these algebra equations gives
*a*_{1},*d*_{1},*a*,*b*,*c*,*d*. Substituting these elements into equation (3.1), the onefold DT is obtained as equation (3.3). By a direct calculation, we know that *H*^{[0]} will be proved in next lemma. ▪

We should note that the most important point to construct the DT is to construct the analytic expression for *H*^{[0]}, so we give the following lemma.

### Lemma 3.2

*With the selection* *then H*^{[0]} *is given by*

### Proof.

Under the selection in remark 2.2, *a*_{1}=*H*^{[0]}, *d*_{1}=1/*H*^{[0]}, *b*=*b*_{0}*H*^{[0]}, *x*-derivative of *f*_{ij}(*i*=1,2). Similarly, using equation (3.5), equation (3.7) and *t*-derivative of *H*^{[0]}
*H*^{[0]} can be expressed as the form in equation (3.8). ▪

### (b) The twofold Darboux transformation

By iteration, the twofold DT for the KE equation is calculated as
*H*^{[1]} possesses the same form as *H*^{[0]} with *f*_{ij} replaced by *i*=1..2,*j*=1..2, i.e.
*T*_{2}(λ)*Φ*_{k}|_{λ=λk}=0(*k*=1…4). Solving these algebraic equations yields
*k*=1,2.

### (c) The *n*-fold Darboux transformation

Let us consider the *n*-fold DT for the KE equation with the similar method as above. By doing *n*-times iteration of the onefold, the DT *T*_{1}, we can obtain *n*-fold DT *T*_{n} of the KE equation, the strategy we have adopted is the inductive method. Now, we briefly elaborate the process. It is trivial to see that we have verified theorem 2.3 for *n*=1,2 from the representation of *T*_{1} and *T*_{2}. Using mathematical induction, we first suppose that the theorem is satisfied when *k*=*n*−1, and then we obtain expression of *H*^{[n−1]}. Next, we just need to prove the theorem for *k*=*n*. Exploiting the iteration of the DT and the representations of *T*_{1} and *T*_{2}, we know that the *n*-fold DT *T*_{n} should be of the form
*T*_{n}(λ,λ_{1},λ_{2},…,λ_{2n−1},λ_{2n})*Φ*_{k}|_{λ=λk}=0, *k*=1,2,…,2*n*. Then, we obtain the elements of *T*_{n} by the Cramer's rule from *T*_{n}*Φ*_{k}|_{λ=λk}=0,*k*=1,2,…,2*n*. Therefore, theorem 2.3 is proved. Furthermore, *T*_{nx}+*T*_{n}*M*=*M*^{[n]}*T*, and we obtain expressions of *u*^{[n]} as defined in theorem 2.4. Note that if *n*=1, *a*^{[1]}=*a*, *b*^{[1]}=*b*, *c*^{[1]}=*c*, *d*^{[1]}=*d* as discussed in lemma 3.1.

## 4. Rogue wave solutions of the Kundu–Eckhaus equation

Now, we consider the breather and RW solution of the KE equation. To begin with, let us assume the seed solution
*k*_{1},*k*_{2} approaches to 1 when *ϵ* goes to zero. Next, with the help of theorems, we construct the breather and RW solution of the KE equation.

### (a) The first-order breather and rogue wave solutions

For *n*=1, let λ_{1}=*ξ*+*η*i, *k*_{1}=1, *k*_{2}=1, *a*=−2*ξ*+2*βc*^{2}, then
*u*^{[1]} is defined by *x*+4*ξt*=0 if *t*-periodic) breather when *ξ*=0. If *u*^{[1]} becomes a temporally periodic (i.e. *x*-periodic) breather solution with a trajectory *t*=0. The profiles of |*u*^{[1]}| are plotted in figure 1.

Furthermore, after a simple analysis, we observe that the periodicity of the breather solution is proportional to 1/*K*_{0}, i.e. when *K*_{0} goes to zero, the distance between the two peaks goes to infinity leaving only one peak located on the (*x*,*t*)-plane. Thus, let *u*^{[1]} in equation (4.4) leads to a new solution with only one local peak which is known as the RW. This kind of solution is shown as
*x* and *t* go to infinity, *η*(=*c*). Moreover, the maximum peak amplitude is equal to 3*η*, which is three times the background amplitude and also satisfies the required definition of RW.

### (b) The higher order rogue wave solutions

Inspired by the above method, we consider the higher order RW solutions of the KE equation in this subsection. However, it is very difficult to obtain higher order RWs from breather solutions, i.e. the explicit expression of *n*th-order breather is very difficult to generate when *n*≥2. Thus, to construct the higher order RWs we directly provide the determinant expression of the solution in the following theorem under the selection in remark 2.2 by higher order Taylor expansion of *u*^{[n]} in theorem 2.4.

### Theorem 4.1

*Let* *and* *by applying the Taylor expansion, then the determinant expression of the nth-order RW is given as
**with
**n*_{i}=2[(i+1)/2]−1, [*i*] *denotes the floor function of i, and*
*Here,* *is the submatrix of* *by deleted the* (2*k*+2)*th column and the* (2*k*+2)*th row*, *is the submatrix of* *by deleted the* (2*k*+1)*th column and the* (2*k*+1)*th row*, *is the submatrix of* *by deleted the* (2*k*+2)*th column and the* (2*k*+1)*th row,* *is the submatrix of* *by deleted the* (2*k*+1)*th column and the* (2*k*+2)*th row. When k*=0, *we define*

There are *n*+2 real parameters (*s*_{i}(*i*=1,2,…,*n*−1),*a*,*c*,*β*) in *n*th-order RW solution. Note that *s*_{i} can be complex numbers, and thus there are (2*n*+1) real constants in the above solutions.

According to theorem 4.1, from the expression of the *n*th-order RW solution, for *n*=1, *k*_{1}=*k*_{2}=1 and there is no non-zero *s*_{i}, we obtain the first-order RW as
*a*=−2*ξ*+2*βc*^{2},*c*=*η*. Note that *c*=−1. The maximum amplitude of *c*, which is obtained under the condition (*x*=0,*t*=0). In order to satisfy *β*=*a*/2*c*^{2}. Under this condition, the first-order RW is parallel to the *t*-axis (figure 6*a*). Four profiles of the first-order RW solutions are plotted in figure 2 for different values of *β*, which shows a significant rotation of these profiles. Correspondingly, parameter *β* affects the phase of the RW. This can be confirmed by changing the real part of the first-order RW Re*β*: the increasing of *β* leads to a strong rotation of the central pattern in figure 3. Note that the asymptotic behaviour of Re *ρ*=0, i.e. *ax*+(−*a*^{2}+4*β*^{2}*c*^{4}+2*c*^{2})*t*=0. Moreover, a solution of the KE equation *u* and a solution of the NLS solution *q* can be connected through a nonlinear transformation *β* on the phase difference is plotted in figure 4, i.e. there exist a remarkable peak and hole around the coordinates in the (*t*,*x*)-plane. If *t* is large sufficiently, *Δθ*=2*βc*^{2}*x* which gives an asymptotic plane. Therefore, the increasing of *β* leads to a rotation of the central pattern of phase difference and also results in the increase of the slope of the asymptotic plane. These observations resemble the signature of the quintic nonlinear and the self-frequency shift effects in the KE equation, and also difference between the NLS equation and the KE equation.

Theorem 4.1 provides a convenient tool to generate different patterns of higher order RWs by suitable choices of *s*_{i}. To save space, we just provide the ring decomposition of the fifth-order RW in figure 5, by set *n*=5 in this theorem. This is the standard decomposition of the RW, which is similar to the ‘wave clusters’ as reported in [40]. In addition, the explicit formulae of second-order and third-order RW solution, and density plots of other higher order RW are given in the electronic supplementary material using theorem 4.1. Effects of real part and imaginary part of *s*_{i} in controlling the distribution of a higher order RW are illustrated graphically in electronic supplementary material, appendix S4.

## 5. Localization characters of the first-order rogue wave solution

In the explicit forms of the RW solutions under fundamental pattern, there are three parameters *a*,*c* and *β*. As we know, it is always a difficult problem to illustrate analytically and clearly the three parameters in the control of the profile for the higher order RW solutions because of the extreme complexity of the solutions. So we only study this problem for the first-order RW solution. Recently, the contour line method has been introduced as an efficient tool to analyse the localization characters of the RW solution in [53]. According to this method, on the background plane with height *c*^{2}, a contour line of *c*=1 is a hyperbola
*k*_{3} be the slope of *l*_{3}, then *k*_{3}=2(*a*−2*β*). There are two fixed vertices: *t*,*x*) plane of all values of *a* and *β*. Here, *l*_{3} is also a median of one triangle composed of above two asymptotes and a parallel one of *x*-axis except *t*=0. We combine the density plots and the above three lines in figure 6 with different values of *β*. At height *c*^{2}+1, a contour line of |*u*_{1rw}|^{2} with *c*=1 is given by a quartic polynomial

According to this formula, we obtain two end points *t*-direction for this closed curve. Moreover, there are two fixed points expressions by *t*,*x*) plane of all values of *a* and *β*. At height *c*^{2}/2, a contour line of *c*=1 is given by a quartic polynomial
*t*. For these two contour lines, there are four fixed points: (0,*t*,*x*) plane of all values of *a* and *β*. Two centres of valley of *a* and *β*. Figure 7 is plotted for above contour lines with different values of

Based on the above analytical results, we could define the length and width of the first-order RW solution. Considering that the contour line on the background of the RW solution is not a closed curve, so we cannot define a length for the first-order RW solution on this plane. Noting that the contour line at height *c*^{2}+*d* is closed, if we set *d* be a positive constant and *c*^{2}+*d*<9*c*^{2}. Without loss of generality, and considering a reasonable height from the asymptotic plane, we can set *d*=1 as before. And we can use the length of the area surrounded by the contour line at height *c*^{2}+1 as the length of the first-order RW solution. Our aim is to find four lines to surround the contour line at height *c*^{2}+1. The length-direction is defined by *l*_{3} in equation (5.3), the width-direction is perpendicular to the length-direction. The reasons for this choice are: (1) *l*_{3} passes through *P*_{3} and *P*_{4}; (2) *l*_{3} is parallel to the tangent line of hyperbola at two vertices ; (3) *l*_{3} is parallel to the tangent line of the contour line at *P*_{5} and *P*_{6}; (4) *l*_{3} is also parallel to two extra outer tangent lines *tl*_{1} and *tl*_{2} of contour line defined by equation (5.4). *tl*_{1} and *tl*_{2} are tangent to the two convex points of contour line. So the length of the first-order RW solution is the distance of *P*_{3} and *P*_{4}, i.e.
*a* and *β*, and can reach minimum length when *a*=2*β*, which can be easily seen in figure 8. Note that the first-order RW solution is parallel to the *t*-axis if *a*=2*β*. In particular, we know from the slope of the *l*_{3}, i.e. *k*_{3}=2(*a*−2*β*), that the increase of *β* results in a significant rotation of the RW solution in the clockwise direction shown in figures 2, 6 and 7. Using formula equation (5.4), we find explicit formulae of *tl*_{1} and *tl*_{2}
*a*≠2*β*. It is trivial to find lines *tl*_{1} and *tl*_{2} are parallel to *t*-axis when *a*=2*β*. Furthermore, there exist two other tangent lines at the two end points *P*_{3} and *P*_{4}, i.e. *x*-axes. So, we obtain four tangent lines *tl*_{1},*tl*_{2},*tl*_{3},*tl*_{4} surrounding the contour line of *c*^{2}+1. The four tangent lines form a parallelogram. Using the expressions of *tl*_{1},*tl*_{2},*tl*_{3},*tl*_{4}, we can solve four vertices *t*,*x*) plane. The width of the first-order RW solution is defined by
*BC*| at width-direction. This formula shows that *d*_{WKE} can reach a maximum when *a*=2*β*, which is plotted in figure 8. The area of the outer tangent parallelogram of the contour line of *c*^{2}+1 is

When *c*=1. The area *S*_{ABCD} provides analytically a good approximation of size of the central red pattern in the density plot of *S*_{ABCD} as an area of the first-order RW solution of the KE equation. This means that the area of the first-order RW is independent of *a* and *β*, which is a surprise result on the localization of the RW solution. Because the length and width of the RW solution are functions of *a* and *β* for a given *c*=1.

In summary, according to our above analytical formulae of slope *k*_{3} along the length direction, the length and width of the first-order RW solution, we find two impacts on localization characters of *β*, such that if *β* is increasing from *a*/2: (1) The length of the RW solution is increasing but the width is decreasing; (2) there exist a significant rotation of the RW along clockwise direction. We also observe oppositely varying trend if *β* is increasing to *a*/2. In optics, *a* is the modulation frequency of the injected light (or equivalently the frequency of the background plane wave) and *β* denotes the quintic nonlinear effect and self-frequency shift effect. Therefore, our results show that one can use them to control the direction and localization characters of the first-order RW in nonlinear optical system governed by the KE equation. Moreover, for a given *c*=1, the area *S*_{ABCD} of the first-order RW solution remains invariant, even for varying *a* and *β*.

## 6. Conclusion and discussion

In this paper, based on the strong physical relevance of non-Kerr (quintic) nonlinear effect and the self-frequency shift effect in a highly nonlinear optical system, we studied a integrable model, i.e. the KE equation as a modelling equation for ultra-short pulse propagation. Although the KE equation [16,17] was introduced 30 years ago as an integrable model from the mathematical point of view, its DT has not been constructed completely as mentioned in the introduction because of the occurrence of the complicated integrals in overall factor *A*_{N} of the reported result in [20]. We overcome that problem by finding an explicit analytical form of the overall factor *n*-fold DT *T*_{n}, and thus got the explicit determinant representation of the *T*_{n} in theorem 2.3 and new solutions *u*^{[n]} in theorem 2.4 for the KE equation. Theorem 2.4 produces new solution *u*^{[n]} of the KE equation. We obtained the *n*th-order RW solutions of the KE equation by *n*-fold DT and higher order Taylor expansion in theorem 4.1. The ring decomposition of the fifth-order RW solutions is plotted according to the analytical formulae in theorem 4.1. In particular, the modulus of the RWs *β*, and this can NOT be generated from a solution *q* of the NLS equation by transformation *β*. Moreover, the phase difference between two first-order RWs of the NLS and the KE has been discussed graphically in figures 3 and 4.

Next, we provided three localization parameters, i.e. length *d*_{LKE}, width *d*_{WKE} and area *S*_{ABCD} for the RW solution of the KE equation. Using the analytical formulae equations (5.6) and (5.7), we proposed two conjectures about localization character of *β*. These aspects have been confirmed visually in figures 2, 6, 7 and 8. Moreover, for a given *c*=1, the area *S*_{ABCD} in equation (5.8) is graphically remain invariant even for varying *a* and *β*.

As the KE equation is a model of ultra-short pulse propagation in a highly nonlinear fibre, our results thus provide the possibility to observe the high-power(ultra-short) RW with the inclusion of non-Kerr (quintic) and self-frequency effects, and also new opportunity to generate high-energy pulse through the higher order RW solutions. As *a* is the modulation frequency of the injected light and *β* denotes the quintic nonlinear effect and self-frequency shift effect, by suitably choosing these parameters, it is possible to use them to control the direction and localization characters of the first-order RW in nonlinear optical system governed by the KE equation.

In addition to the first-order breather, our representation of the DT can also be used to generate the higher order breather of the KE equation, and the latter is necessary to study relationship between higher order RW solutions and breathers. For example, by using the limiting condition of degenerate eigenvalues through higher order Taylor expansion, a degenerate higher order breather generates a higher order RW solution. This observation has been supported approximately and numerically by plotting profile in the interaction region of three breathers, which looks like a third-order RW very much, for the NLS equation [43] (see figures 2 and 3 of this reference). Here, we plot two second-order breathers in figure 10 for the KE equation to confirm this observation again, and the detailed research of this work will be published elsewhere. Moreover, if we do not set *c*=1 and *d*=1 to consider the contour line of

## Note added in proof

After we finished this paper, we came to know about the recently published paper (*Phys. Scr*. **89** (2014) 095210) has constructed RW solutions of the KE equation up to fourth-order. The authors of this paper solve an integro-differential equation (which is called an extended NLS, see eqn (3) of this paper) and then get a solution *u* of the KE from a solution *q* of the extended NLS equation by a transformation

## Competing interests

We declare we have no competing interests.

## Authors' contributions

D.Q., J.H. and Y.Z. constructed the DT and solutions. J.H. and K.P. provided physical concerns of KE equation and localized analysis of RW, and wrote the paper. All authors gave final approval for publication.

## Funding

This work is supported by the NSF of China under grant no. 11271210, the K. C. Wong Magna Fund in Ningbo University. K.P. thanks the DST, NBHM, IFCPAR, DST-FCT and CSIR, Government of India, for the financial support through major projects.

## Acknowledgements

J.S.H. acknowledges sincerely Prof. A. S. Fokas for arranging the visit to Cambridge University in 2012–2014 and for many useful discussions. K.P. thank Prof. Philippe Grelu and Prof. T. Patrice Dinda for their kind Hospitality at Universite of Bourgogne, Dijon. We thank the editorial member and referees for useful suggestions on the first submission at 16 October 2014 and the second submission at 10 April 2015.

- Received April 10, 2015.
- Accepted June 30, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.