Periodic wave solutions of nonlinear equations by Hirota's bilinear method

Based on a Riemann theta function and Hirota's bilinear form, a lucid and straightforward way is presented to explicitly construct double periodic wave solutions for both nonlinear differential and difference equations. Once such a equation is written in a bilinear form, its periodic wave solutions can be directly obtained by using an unified theta function formula. The relations between the periodic wave solutions and soliton solutions are rigorously established. The efficie...

It is shown that, by letting wavenumbers and frequencies complex in Hirota's bilinear method, new classes of exact solutions of soliton equations can be obtained systematically. They include not only singular or N-homoclinic solutions but also N-wavepacket solutions.

In this work, we focus on the construction of Nth-rouge wave solutions for the Hirota equation by utilizing the bilinear method. The formula can be represented in terms of determinants. In addition, some interesting dynamic patterns of rogue waves are exhibited.

A series of new soliton solutions are presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann Hilbert method and transformation relationship. First, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special transformation relat...

We give an elementary introduction to Hirota's direct method of constructing multisoliton solutions to integrable nonlinear evolution equations. We discuss in detail how this works for equations in the Korteweg-de Vries class. We also show how Hirota's method can be used to search for new integrable evolution equations by testing for the existence of 3- and 4-soliton solutions, and list the results that have been obtained this way for the KdV, mKdV/sG and nlS classes of equat...

The Hirota bilinear formalism and soliton solutions for a generalized Volterra system is presented. Also, starting from the soliton solutions, we obtain a class of nonsingular rational solutions using the "long wave" limit procedure of Ablowitz and Satsuma, and appropriate "gauge" transformations. Their properties are also discussed and it is shown that these solutions interact elastically with no phase shift.

This article is devoted to exact solutions of the Boussinesq equation that models nonlinear shallow water waves. For this we use the Hirota bilinear method and differential constrains. Out solutions describe in particular the motion of the wave packets, waves on soliton and "dancing" waves. We present a simple method for multiplication of solutions of the Hirota equation which allow us to generate more complex structures from the waves.

We consider coupled nonlinear Schrodinger equations (CNLSE) which govern the propagation of nonlinear waves in bimodal optical fibers. The nonlinear transform of a dual-frequency signal is used to generate an ultra-short-pulse train. To predict the energy and width of pulses in the train, we derive three new types of travelling periodic-wave solutions, using the Hirota bilinear method. We also show that all the previously reported periodic wave solutions of CNLSE can be deriv...

Bilinearization of a given nonlinear partial differential equation is very important not only to find soliton solutions but also to obtain other solutions such as the complexitons, positons, negatons, and lump solutions. In this work we study the bilinearization of nonlinear partial differential equations in $(2+1)$-dimensions. We write the most general sixth order Hirota bilinear form in $(2+1)$-dimensions and give the associated nonlinear partial differential equations for ...

In a previous work[1] exact stable oblique soliton solutions were revealed in two dimensional nonlinear Schroedinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is "close" to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit.