Yulei Cao(曹玉雷), Peng-Yan Hu(胡鵬彥), Yi Cheng(程藝), and Jingsong He(賀勁松)

1Institute for Advanced Study,Shenzhen University,Shenzhen 518060,China

2College of Mathematics and Statistics,Shenzhen University,Shenzhen 518060,China

3School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China

Keywords: two-dimensional (2D) Korteweg-de Vries (KdV) equation, Bilinear method, B¨acklund transformation,Lax pair,deformed 2D rogue wave

1. Introduction

Rogue waves (RWs) in the ocean, also known as killer waves, monster waves, freak waves, and extreme waves, are special waves with extremely destructive,which should be responsible for some maritime disasters. The main feature of RW is that “appear from nowhere and disappear without a trace”.[1]Peregrine was the first one who obtained the RW solution from one-dimensional systems,[2]so such RW solution is also called “Peregrine soliton”. These RWs are localized in both space and time in one-dimensional systems,[3-9]and its dynamic behaviors are similar to that of lumps in highdimensional systems.[10-14]RWs in high-dimensional systems are often called line RWs, which are merely localized in time.[15-18]The study of RW has been widely used in diverse areas of theoretical and applied physics, including plasma physics,[19,20]Bose-Einstein condensates,[21,22]atmosphere physics,[23]optics and photonics,[24-26]and superfluids.[27]

Now, a fundamental problem can be asked: Is it possible that other types of RW solutions exist in nonlinear systems? Seeking new exact solutions for nonlinear systems is an open problem and a challenging work. Furthermore, in integrable systems, the Peregrine soliton only appears in the one-dimensional systems. But, the RWs, which are localized in time and space, have never been found in the highdimensional system. Therefore,searching for the RW solution that is local in time and space in high-dimensional systems has always been an open problem. Until recently, Guo et al.[28]obtained RW solutions,which are localized in both space and time,from a two-dimensional nonlinear Schr¨odinger model by the even-fold Darboux transformation. Now the long-standing problem of the construction of two-dimensional (2D) rogue wave on zero background has been addressed.[28]This is a pioneering work, which inspires us to explore the structure of 2D RW solutions of other high-dimensional soliton equations. Additionally, the RW solutions of complex nonlinear equations have been widely researched,but the RW solutions of real nonlinear equations are rarely mentioned,which is another motivation.

Inspired by the above considerations, in this paper, we focus on the (2+1)-dimensional Korteweg-de Vries (KdV)equation

which was first derived by Boiti-Leon-Manna-Pempinelli with a weak Lax pair,[29]and can be reduced to the celebrated KdV equation if x=y. Equation (1) is an asymmetric part of the Nizhnik-Novikov-Veselov(NNV)equation,[30]thus equation(1)is also called ANNV equation;this equation can also be derived by using the inner parameter-dependent symmetry constraint of the KP equation,[30]and can be considered as a model of an incompressible fluid where U and V are the components of velocity.[31]Equation (1) was also regarded as a generalization[32]of the results of Hirota and Satsuma[33]in the(2+1)dimensions.

Let U =Wy,V =Wx,then equation(1)reduces to the following single equation:

Equation (2) known as BLMP equation was widely investigated. The Cauchy problem, associated with initial data decaying sufficiently rapidly at infinity, was solved by means of inverse scattering transformation.[29]The spectral transformation, nonclassical symmetries, and Painlev′e property for Eq. (2) have been studied in Refs. [29,34,35]. And a series of B¨acklund transformations,[36-38]Lax pairs,[38]supersymmetric,[39]soliton-like,[40-43]breather,[44,45]and lump-type[46-48]solutions were derived.

In this paper, we are committed to exploring new B¨acklund transformations, Lax pairs, and new RW solutions of 2D KdV equation(1)by the Hirota method. The new RW solution is called deformed 2D RW because its formula involves an arbitrary function φ(y) which provides a deformed background for the rogue wave. This arbitrary function φ(y)comes from the crucial bilinear transformation which maps U to a bilinear form.In Section 2,new B¨acklund transformations and Lax pair are derived based on the binary Bell polynomials and linearizing the bilinear equation. In Section 3, we obtain the deformed kink soliton and deformed breather solutions of the 2D KdV equation by means of the Hirota method. In Section 4,a family of new rational solutions and deformed 2D RW solutions of Eq. (1) are presented by introducing an arbitrary function φ(y). The main results of the paper are summarized and discussed in Section 5.

2. Bilinear Ba¨cklund transformation and associated Lax pair

In this section, we mainly focus on the new B¨acklund transformation and Lax pair of 2D KdV equation (1).B¨acklund transformation and lax pair are recognized as the main characteristics of integrability,which can be used to obtain exact solutions of nonlinear systems. We first introduce the following variables transformation:

then 2D KdV equation(1)becomes

We introduce two new variables

Based on the binary Bell polynomials, equation (5) can be rewritten into y-polynomials from Bell polynomial theory as follows(see Refs.[51-53]for details):

We have the following constraints:

where λ is the constant parameter. Then, the following coupled system of y-polynomials of Eq.(1)is deduced:

According to the relationship between y-polynomials[51-53]and bilinear operators D[54]

the following bilinear B¨acklund transformation is derived for the 2D KdV equation(1):

Then logarithmic linearization of y-polynomials under the Hopf-Cole transformation v=lnψ,and Bell polynomial formula

the y-polynomials can be written as

thus system(9)is then linearized into a Lax pair

which is equivalent to the Lax pair of Eq.(1)[36]

ψxy?Uψ ?λψx=0,

In order to get a new B¨acklund transformation, we introduce the following gauge transformation to the bilinear B¨acklund transformation(11):

where ξ = m1x+n1y+l1t and η = m2x+n2y+l2t. System(11)is transformed into

under the following constraints:

Then,a new B¨acklund transformation of Eq.(1)is derived as follows:

Similarly,the Lax pair of system(1)is as follows:

or equivalently

It is easy to check that the integrability condition[L1?U,?t+L2]ψ =0 is satisfied.

Additionally, another form of B¨acklund transformation and lax pair of 2D KdV equation (1) are also derived, when taking

2D KdV equation(1)is then translated into the following bilinear form:

where φ(y) is an arbitrary function of y, f is a real function,and D is the Hirota’s bilinear differential operator,[54]based on the binary Bell polynomials,[49-51]equation(1)admits the following B¨acklund transformation:

If using gauge transformation (16) to the above bilinear B¨acklund transformation(24),a new B¨acklund transformation of Eq.(1)is derived

Then, we derive the corresponding linear system under the Hopf-Cole transformation(13)

which is equivalent to the Lax system of 2D KdV equation(1)

3. Deformed multi-kink soliton and deformed breather solutions of the 2D KdV equation

Based on the above new bilinear equation(23),some exact solutions of 2D KdV equation (1), including deformed multi-solitons and deformed breathers, are generated. First,the deformed one-soliton solutions take the forms

The exact expressions of U[1]and V[1]are as follows:

Fig.1. Deformed one-soliton U[1] and V[1] of Eq.(1)in the(x,y)-plane with parameters p1=1,q1=?4 and displayed at t=0.

In order to obtain the deformed two-solitons of 2D KdV equation(1),we take

where

The dynamic behaviors of the deformed two-soliton solutions are more complex and interesting by choosing the appropriate parameter φ(y) see Fig.2. Furthermore, as can be seen in the two-dimensional plots of Fig.2, the interaction of the deformed two-soliton solutions is an elastic collision.

Similarly, N-soliton solutions U[N]and V[N]are given in Eq.(22)of the 2D KdV equation(1),in which f can be written as follows:

with

Fig.2. Deformed two-soliton solutions U[2] and V[2] of Eq. (1) in the (x,y)-plane displayed at t =0. Panels (a)-(d) are the two-dimensional plots of(e)-(h),respectively.

4. Deformed 2D RW of the 2D KdV equation

In order to obtain the deformed 2D RW solutions of the 2D KdV equation(1),taking

in Eq.(30)and a suitable limit as p1, p2→0,then we further take λ1=a+bi,λ2=a ?bi in Eq. (36). The solutions U[1]and V[1]are given as follows:

with

As can be seen from the above expression, to ensure that the above solutionsU[1]andV[1]are smooth,parameter a ＞0 must be held.

4.1. Fundamental rational solutions

The fundamental rational solutions U[1]and V[1]of 2D KdV equation(1)are derived if φ(y)is a polynomial function.Without loss of generality,take φ(y)=2y. The trajectories of U[1]and V[1]are as follows:

However, the dynamic behavior of the rational solution V[1]is different from that of U[1]. For given φ(y)=2y,the solution V[1]has the following nine critical points in(x,y)-plane:

letting

yields

Based on the above analysis,the evolution of rational solution V[1]of 2D KdV equation(1)can be divided into the following three stages.

Fig.3. The temporal evolution of fundamental rational solution U[1] removing the background plane φ(y) of Eq. (1) in the (x,y)-plane with parameters φ(y)=2y,a=2,and b=2. Panels(d)and(e)are two-dimensional plots of U[1].

Obviously,the global extreme values of rational solution V[1]change with time,and

(ii) When t =0, rational solution V[1]has three extreme points

The maximum and minimum values are as follows:

From the above analysis,we can seen that its dynamic behavior is similar to the RW in one-dimensional systems. The amplitude of the RW in one-dimensional systems is three times that of the background plane. However,the amplitude of RW of 2D KdV equation(1)is controlled by parameters a and b.

(iii) When t ＞0, rational solution V[1]has six extreme points Λ4,5and Λ6,7,8,9. And it has four maximums and two minimums

Obviously,the extreme values of V[1]at t=0 are equal to the extreme values at t ＞0.

Fig.4. The temporal evolution of fundamental rational solution V[1] of Eq.(1)in the(x,y)-plane with parameters φ(y)=2y,a=2,and b=2.

4.2. Fundamental deformed 2D RW solutions

As can be seen in Fig.5,in order to better observe the evolution of the deformed 2D RW U[1],we remove the background plane of U[1]. Four panels describe the appearance and annihilation of 2D RW in(x,y)-plane along the curve x+2sech(y).This is the first time to obtain such deformed 2D RW in highdimensional systems. The dynamic behavior of fundamental deformed 2D RW V[1]is more complicated and interesting.

Fig.5.The temporal evolution of fundamental deformed 2D RW solution U[1]removing the background plane φ(y)of Eq.(1)in the(x,y)-plane with parameters Φ(y)=sech(y),a=1,and b=4.

Through simple calculation and analysis,the evolution of deformed 2D RW V[1]can be divided into the following four stages

(i)When t ＜0,a line RW appears from the constant background plane,with the time evolution,the amplitude of the line RW gradually increases.

Fig.6. The temporal evolution of fundamental deformed 2D RW solution V[1] of Eq.(1)in the(x,y)-plane with parameters Φ(y)=sech(y),a=1,and b=4.

4.3. High-order deformed 2D RW solutions

Take

in Eq.(34). Further taking

As can be seen in Fig.7, by removing the background plane φ(y), the second-order deformed 2D RW solution U[2]describes that four Peregrine-type solitons appear and annihilate from the constant background plane. However, the dynamic behaviors of the second-order RW solution V[2]are similar to that of the fundamental RW solution V[1]. The RW solution V[2]uniformly approaches to a constant background plane when t →±∞. Two line RWs appear from the constant plane under the time evolution,and their amplitudes increase rapidly.Then the amplitudes of the two line RW attenuate rapidly, at the same time, two Peregrine-type solitons are produced. Finally, these two Peregrine-type solitons disappear in a very short time without a trace(see Fig.8).

Fig.7. The temporal evolution of second-order deformed 2D RW solution U[2] removing the background plane φ(y)of Eq.(1)in the(x,y)-plane.

Fig.8. The temporal evolution of second-order deformed 2D RW solution V[2] of Eq.(1)in the(x,y)-plane.

For larger N,higher-order solutions are generated by taking the parameters

in Eq.(34)and a suitable long wave limit as pj→0,the function f defined in Eq. (34) becomes a polynomial-type function containing an arbitrary function φ(y). Therefore,the general n-th rational-type functions U[n]=?2ln fxy+φ(y) and V[n]=?2ln fxxof 2D KdV equation(1)can be derived,[55]in which f can be written as follows:

with

Remark 1The above solutions U[n]and V[n]are rational solutions, if φ(y) is a nonzero polynomial function. For example,when φ(y)=2y,the rational solution U[n]describes the fission of n-bright lumps and n-dark lumps from the background plane 2y. The rational solution V[n]describes the fission of 2n-dark lumps from a constant background plane.

Remark 2The above solutions U[n]and V[n]are deformed 2D RW solutions if Φ(y)=sech(y). The solution U[n]describes 2n Peregrine-type solitons that appear and annihilate from a kink-soliton plan. The solution V[n]shows that n-line RWs appear and decay rapidly from a constant background plane, and fission into n Peregrine-type solitons, and finally disappear in the constant background plane without a trace.

5. Summary

In this paper, new bilinear B¨acklund transformation and Lax pair of the 2D KdV equation (1) are derived, which are different from the B¨acklund transform and Lax pair in Refs.[36,37]. Teh N-soliton solutions are presented by means of the improved Hirota’s bilinear method. Deformed soliton and deformed breather solutions of elastic collision are generated by selecting the appropriate free parameter φ(y), see Figs.1 and 2. When Φ(y)is a non-constant polynomial function, a family of new rational solutions of the 2D KdV equation are generated using the long wave limit. When Φ(y) ia a non-zero constant, the rational solutions U[n]and V[n]reduce to the rational solution of 2D KdV equation in Ref.[47].When Φ(y)=sech(y), the deformed 2D RW solution U describes a family of Peregrine-type solitons that appear and annihilate from a kink-soliton plan.In order to better observe the evolution of the deformed 2D RW solutions, we remove the background plane of U when plotting,see Figs.5 and 7. The deformed 2D RW solution V shows that a series of line RWs appear and decay rapidly from a constant background plane,and fission into Peregrine-type solitons,and finally annihilate in the constant background plane without a trace, see Figs. 6 and 8. This paper successfully constructed the deformed 2D RW solutions,which are closely related to the introduced arbitrary function Φ(y). These novel phenomena have never been reported before in nonlinear systems. Our presented work not only provides a new reference method for seeking new exact solutions of nonlinear partial differential equations, but also may be helpful to promote a deeper understanding of nonlinear phenomena.