Double and triple poles solutions for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions (2104.12073v2)

Abstract: In this work, the double and triple poles soliton solutions for the Gerdjikov-Ivanov(GI) type of derivative nonlinear Schrödinger equation with zero boundary conditions(ZBCs) and nonzero boundary conditions(NZBCs) are studied via Riemann-Hilbert (RH) method. Though spectral problem analysis, we first give out the Jost function and scattering matrix under ZBCs and NZBCs. Then according to the analyticity, symmetry and asymptotic behavior of Jost function and scattering matrix, the Riemann-Hilbert problem(RHP) with ZBCs and NZBCs are constructed. Further, the obtained RHP with ZBCs and NZBCs can be solved in the case that reflection coefficients have double or triple poles. Finally, we derive the general precise formulae of N-double and N-triple poles solutions corresponding to ZBCs and NZBCs, respectively. The dynamical behaviors for these solutions are further discussed by image simulation.

• The paper presents novel N-double and N-triple pole solutions for the GI equation using a refined Riemann-Hilbert approach.
• It demonstrates distinct soliton interactions under zero boundary conditions and breather interactions under nonzero ones.
• The study extends integrable system theory by introducing methods to regularize spectral singularities in nonlinear wave dynamics.

Double and Triple Poles Solutions for the Gerdjikov-Ivanov Type of Derivative Nonlinear Schrödinger Equation

Introduction

The paper presented in the paper "Double and triple poles solutions for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions" (2104.12073) investigates the solutions for a specific form of the derivative nonlinear Schrödinger equation known as the Gerdjikov-Ivanov (GI) equation. The GI equation is represented by:

$$iu_{t} + u_{xx} - iu^{2}u_{x}^{\ast} + \frac{1}{2}|u|^{4}u = 0,$$

where $u$ denotes a transverse magnetic field perturbation function, with $x$ and $t$ representing spatial and temporal variables, respectively. This equation is significant in modeling Alfve'n waves in plasma physics.

The paper aims to construct solutions characterized by double and triple poles for this equation under both zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs). This is accomplished using the sophisticated Riemann-Hilbert (RH) method.

Riemann-Hilbert Problem Construction

Zero Boundary Conditions:

For ZBCs, where $u(x, t) \rightarrow 0$ as $x \rightarrow \pm \infty$, the paper establishes a formal RH problem by first analyzing the spectral problem, constructing Jost functions and deriving the scattering matrix. The RH problem is then constructed based on the analyticity, symmetry, and asymptotic behavior of these functions. The double and triple poles solutions are obtained by regularization of the non-regular RH problem through singularity elimination.

Nonzero Boundary Conditions:

For NZBCs, where $$u(x, t) \rightarrow u_{\pm} e^{-\frac{3}{2}iu_{0}^{4}t+iu_{0}^{2}x}$$, the RH problem is expanded to accommodate more complex boundary conditions. This involves mapping the problem onto a $z$-plane instead of a Riemann surface for effective spectral analysis, followed by constructing the RH problem with adjusted jump matrices.

Solutions and Results

Double Poles Solutions:

The paper introduces $N$-double pole solutions under both ZBCs and NZBCs. These solutions are derived from the RH problem by considering reflectless conditions (setting reflection coefficients to zero) and applying matrix decomposition techniques. Solutions manifest as soliton interactions that are computationally represented and dynamically analyzed for various initial conditions. Under ZBCs, these solutions illustrate interactions between bright solitons, while under NZBCs, they imply interactions between breather waves and solitons.

Triple Poles Solutions:

Similarly, $N$-triple pole solutions are derived, representing complex interactions such as bright-bright-bright solitons under ZBCs and breather-breather-breather interactions under NZBCs. The solutions involve higher-order pole techniques and provide insight into multi-soliton dynamics.

Implications

The methods and solutions proposed in the paper extend the theoretical and practical understanding of integrable systems in mathematical physics, specifically for nonlinear wave equations. They offer novel approaches for modeling complex wave interactions in physical systems, including plasma physics and optics. The RH method demonstrated provides robust tools for addressing non-standard boundary conditions and spectral singularities, potentially influencing further studies in integrable models with similar properties.

Conclusion

This research enhances computational soliton theory by delivering new types of solutions for the GI equation via the RH approach, paving the way for advanced studies into nonlinear wave dynamics and offering methodologies applicable to broader classes of integrable equations. Further exploration on regularization techniques might unearth more complex soliton behaviors in nonlinear systems governed by higher-order poles.