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

We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula. We consider two types of double periodic solutions as seed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darboux transformation formula through an algebraic method with two eigenvalues. We then construct the desired solution in two steps. In the first step, we create double periodic waves a...

The goal of this work is to determine whole classes of solitary wave solutions general for wave equations.

The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.

The Hirota equation is better than the nonlinear Schr\"{o}dinger equation when approximating deep ocean waves. In this paper, high-order rational solutions for the Hirota equation are constructed based on the parameterized Darboux transformation. Several types of this kind of solutions are classified by their structures.

Under investigation in this work is the coupled Hirota system arising in nonlinear fiber. The spectral analysis of the Lax pair is first carried out and a Riemann-Hilbert problem is described. Then in the framework of the obtained Riemann-Hilbert problem with the reflectionless case, N-soliton solution is presented for the coupled Hirota system. Finally, via proper choices of the involved parameters, a few plots are made to exhibit the localized structures and dynamic charact...

The influence of fractional order parameter $(\alpha)$ in nonlinear waves is examined in the fractional Zakharov-Kuznetsov (FZK) equation with the Hirota bilinear approach. Symbolic computation is used for all mathematical calculations. A significant impact of the fractional order parameter is found on the single and multi-soliton solutions. The fact that the structural change is noticeable when $\alpha$ is raised, is crucial to our investigation.

Kadomtsev-Petviashvili (KP) equation, who can describe different models in fluids and plasmas, has drawn investigation for its solitonic solutions with various methods. In this paper, we focus on the periodic parabola solitons for the (2+1) dimensional nonautonomous KP equations where the necessary constraints of the parameters are figured out. With Painleve analysis and Hirota bilinear method, we find that the solution has six undetermined parameters as well as analyze the f...

Considering the coupled envelope equations in nonlinear couplers, the question of integrability is attempted. It is explicitly shown that Hirota's bilinear method is one of the simple and alternative techniques to Painlev\'e analysis to obtain the integrability conditions of the coupled nonlinear Schr\"odinger (CNLS) type equations. We also show that the coupled Hirota equation introduced by Tasgal and Potasek is the next hierarchy of the inverse scattering solvable CNLS equa...

Under investigation in this work is the robust inverse scattering transform of the discrete Hirota equation with nonzero boundary conditions, which is applied to solve simultaneously arbitrary-order poles on the branch points and spectral singularities. Using the inverse scattering transform method, we construct the Darboux transformation but not with the limit progress, which is more convenient than before. Several kinds of rational solutions are derived in detail. These sol...

Physically relevant soliton solutions of the resonant nonlinear Schrodinger (RNLS) equation with nontrivial boundary conditions, recently proposed for description of uniaxial waves in a cold collisionless plasma, are considered in the Hirota bilinear approach. By the Madelung representation, the model is transformed to the reaction-diffusion analog of the NLS equation for which the bilinear representation, soliton solutions and their mutual interactions are studied.