The rogue waves and their dynamics in the Hirota—Maccari equation

Zhai Wen-Yan

Abstract： In this paper， general high-order rogue waves in the Hirota-Maccari equation are derived by the Hirota bilinear method. By setting the regulation of free parameters， the presentation of fundamental rogue waves are demonstrated by the figures and density.

Key words： Hirota-Maccari equation； soliton； rogue waves； bilinear method

1 Introduction

Rogue waves have became the subject of intensive investigations in a majority of physical systems. The methods of constructing rogue wave solutions for integrable system are abundant. Especially， the generalized Darboux transformation method and bilinear method. The multi-rogue waves or higher-rogue waves in high dimensional equation are interesting in realistic problems because of the rogue wave solutions in these equations have more complex changes.

As a generalized Hirota equation，（2+1）-dimensional Hirota-Maccari （HM） system was derived by Maccari. Our goal in this paper is to find the rogue waves of the HM equations that are expressed in terms of Grammian determinants based on the Hirota's bilinear method and the reduction technique.

2 Derivation of general rogue-wave solutions

We consider the Hirota-Maccari equation in the following form

（1）

where u and v are complexes.

Now we make coordinate transformation Eq.（1） become

（3）

Using the independent variable transformation

where f is a real function and g is a complex one with respect to variables x，y and t. Then we can see that Eq.（3） is transformed into the following bilinear form

（5）

where '-' represents complex conjugation . Nth-order rational solution of Eq.（3） can be constructed by the following theorems.

Theorem. The Eq.（3） has rational solutions （4） with f and g are given by N×N determinant.

（6）

where

and the matrix elements are given by

where

（8）

Here，ak（k=1…，N） are the arbitrary complex contents and i，j are positive integers.

3 Dynamics of the rogue waves

Setting N=1 in Theorem， and set a1 to zero ，we produce the first order rogue wave

where

Fig.1 Fundamental rogue wave|u1|in （9） with ？茁=1，t=0

Fig.1shows the fundamental rogue wave of |u1| in Eq.（9）， its maximum peak amplitude is equal to 3.

4 Conclusion

It has been shown that the fundamental rogue wave and multi-rogue waves of（2+1）-dimensional HM equation can be produced with the use of bilinear method. These results demonstrate that bilinear method is a powerful tool to construct rogue waves of nonlinear evolution equations.

References

[1]Ohta Y， Yang JK. Proceedings of the Royal Society of London Series A 2012；468（2142）：1716-1740.

[2]Hirota R. J Math Phys 1973；14（7）：805-809.

Corresponding author：Zhai wenyan.