Zhao Zhang,Junchao Chen and Qi Guo

1 Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices,South China Normal University,Guangzhou 510631,China

2 Department of Mathematics and Institute of Nonlinear Analysis,Lishui University,Lishui 323000,China

Abstract Based on the Hirota’s method,the multiple-pole solutions of the focusing Schr?dinger equation are derived directly by introducing some new ingenious limit methods.We have carefully investigated these multi-pole solutions from three perspectives:rigorous mathematical expressions,vivid images,and asymptotic behavior.Moreover,there are two kinds of interactions between multiple-pole solutions:when two multiple-pole solutions have different velocities,they will collide for a short time; when two multiple-pole solutions have very close velocities,a long time coupling will occur.The last important point is that this method of obtaining multiple-pole solutions can also be used to derive the degeneration of N-breather solutions.The method mentioned in this paper can be extended to the derivative Schr?dinger equation,Sine-Gorden equation,mKdV equation and so on.

Keywords:multiple-pole solutions,degenerate solutions,Hirota’s bilinear method

1.Introduction

Nonlinear partial differential equations are often used to model various phenomena in fields such as physics,chemistry,biology and even social sciences[1–11].With the help of exact solutions,various nonlinear phenomena can be better explained[1–7].The construction of exact solutions of nonlinear equations,especially soliton solutions[12–14],is one of the most important and essential tasks in nonlinear science.

It is well known that the famous focusing nonlinear Schr?dinger equation(FNLS equation for short)

can be used to describe optical solitons,self-trapping phenomena of nonlinear optics,propagation of thermal pulses in solids,Langnui waves in plasma,the motion of superconducting electrons in electromagnetic fields,Bose–Einstein condensation effect of atoms in lasers,and so on[15–17].

There have been hundreds of studies and monographs on how to find solutions to equation(1)[18–34].N-soliton solutions,breather solutions,rogue wave solutions,and multiple-pole solutions of the FNLS equation have been derived by Darboux transformation,inverse scattering and other approaches[23–30].In particular,a very mature mechanism has been developed for obtaining multiple-pole solutions,higher-order rogue wave solutions and rogue waves with a double-periodic background using the Darboux transformation[25–30].

The multiple-pole solution is actually a weakly bound state of solitons,where the velocities and amplitudes of the solitons are almost the same.In this weakly bound state,the soliton travels along a curve,and the velocity and amplitude tend to be fixed values when the time is large enough.As with the multiple-pole solutions,similar asymptotic behavior exists for degeneration of breather solutions.As early as last century,it was reported that multiple-pole solutions of the FNLS equation were obtained by the inverse scattering method[31,32].In 2017,Schiebold[23]produced an important work that gave a rigorous and complete asymptotic description of the multiple-pole solutions for the FNLS equation.In the same year,Wanget al[29]first discovered the degeneration ofN-breather solutions using the generalized Darboux transformation and creatively pointed out the connection between it and higher-order rogue waves.Subsequently,some scholars linked the inverse scattering method[24]or the Darboux transformation method[33]to the Riemann–Hilbert problem to explore the higher-order multiple-pole solutions.

In short,most of the research to obtain the multiple-pole solution still use the inverse scattering method[23,24,31,32]and Darboux transformation method[33,34]which have higher requirements on mathematics.

However,there is no report on whether the Hirota’s method[14,35–40],as a direct method to obtain the exact solution of the integrable system,can derive the multiple-pole solution of FNLS equation.According to the bilinear method,theN-soliton solution of the FNLS equation takes the following form[19,20,41]:

The latest research[42]mentions a way to derive multiple-pole solutions directly fromN-soliton solutions,which provides a possibility to concisely derive degenerate solutions from equation(2).After tedious verification,the idea mentioned in[42]is feasible for the FNLS equation.In other words,the multiple-pole solutions of equation(1)can be obtained by the Hirota’s bilinear method.Similarly,degenerate solutions can be derived directly fromN-breather solutions without too much esoteric mathematics using these skillful limit tricks.

2.Double-pole solutions

Figure 1.(a)∣u2-p∣described by equation(4)with parameters ;(b)Contour plot of(a),and the cyan curve is the trajectory of the wave crest predicted by equation(7).

Figure 2.Two different types of interactions between double-pole solutions:(a)a brief collision described by proposition 2.2 with parameters (b)a strong coupling described by proposition 2.2 with parameters.

More generally,by setting

in equation(2),then a 2M-soliton solution will be reduced to an interaction betweenMdouble-pole solutions when ?→0.

Regretfully,the computational difficulty of theN-soliton solution of the FNLS equation is equivalent to that of the 2Nsoliton solution of the mKdV equation[42].Therefore,it is difficult to derive general expressions for the interaction ofMdouble-pole solutions.However,the Darboux transformation method which requires a higher mathematical foundation does not have this defect[43].

3.Multiple-pole solutions

Figure 3.(a)∣u3-p∣described by equation(11)with parameters ;(b)Contour plot of(a),and the cyan curve represents the trajectory predicted by equation(14).

4.Degeneration of N-breather solutions

Figure 4.(a)∣u 4-p∣described by proposition 3.2 with parameters ;(b)Contour plot of(a),and the cyan curve represents the trajectory predicted by equation(16).

Figure 5.(a)A degenerate solution of a 2-breather solution∣u-2db∣described by equation(20);(b)by selecting parameters in proposition 4.2,a special degenerate solution|u3?db|can be derived from a 3-breather solution.

5.Conclusion

Based on the bilinear method,this paper systematically investigates the multiple-pole solutions and degenerate solutions of focusing nonlinear Schr?dinger equation by some skillful limiting means.The greatest innovation of this study is that it provides a simple and fast method to derive the multiple-pole solutions of FNLS equation.Proposition 2.1 and proposition 3.1 give specific approaches to the doublepole solution and the triple-pole solution and give general expressions equation(4)and equation(11)for the relevant solutions.Proposition 3.2 summarizes proposition 2.1 and proposition 3.1,and points out a general method to derive multiple-pole solutions directly fromN-soliton solutions.The space-time structure and dynamical properties of these multipole solutions are described by beautiful images(figures 1,3,4)and rigorous mathematical expressions(equations(7),(14),(16))respectively.Similarly,this limit method mentioned in this paper can also be used to achieve degeneration of aNbreather solution(see proposition 4.1 and proposition 4.2 for details).There are two limiting steps to convert the higherorder breather solution to the higher-order rogue wave solution.For the first limiting step,proposition 4.1 has been able to derive a degenerate solution from theN-breather solution;however,for the second step,it is worth further thinking how to convert the higher-order degenerate solution obtained at the first stage into a higher-order rogue wave.In addition,this method can be extended to a derivative Schr?dinger equation,sine-Gorden equation,mKdV equation and so on.

Acknowledgments

This research is supported by the Natural Science Foundation of Guangdong Province of China(No.2021A1515012214),the Science and Technology Program of Guangzhou(No.2 019 050 001),National Natural Science Foundation of China(Nos.12 175 111),and K C Wong Magna Fund in Ningbo University.The authors sincerely thank Dr Jiguang Rao(Shenzhen University)for his suggestions and encouragement.

Declarations

Conflict of interest

The authors declare that they have no conflict of interests.