[Paper Review] General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear equations

Bo Yang, Yong Chen|arXiv (Cornell University)|Dec 16, 2017

Nonlinear Waves and Solitons3 citations

TL;DR

This paper constructs general rogue wave solutions for reverse time integrable nonlocal nonlinear equations—specifically the nonlocal NLS and Davey-Stewartson (DS) equations—using a unified binary Darboux transformation (DT) method. It reveals that rogue waves can exhibit bounded dynamics or finite-time singularities depending on free parameters, with higher-order solutions showing novel hybrid structures like parabolic rogue waves and reflecting lumps, which are absent in local counterparts.

ABSTRACT

A study of general rogue waves in some integrable reverse time nonlocal nonlinear equations is presented. Specifically, the reverse time nonlocal nonlinear Schrödinger (NLS) and nonlocal Davey-Stewartson (DS) equations are investigated, which are nonlocal reductions from the AKNS hierarchy. By using Darboux transformation (DT) method, several types of rogue waves are constructed. Especially, a unified binary DT is found for this nonlocal DS system, thus the solution formulas for nonlocal DSI and DSII equation can be written in an uniform expression. Dynamics of these rogue waves is separately explored. It is shown that the (1+1)-dimensional rogue waves in nonlocal NLS equation can be bounded for both x and t, or develop collapsing singularities. It is also shown that the (1+2)-dimensional line rogue waves in the nonlocal DS equations can be bounded for all space and time, or have finite-time blowing-ups. All these types depend on the values of free parameters introduced in the solution. In addition, the dynamics patterns in the multi- and higher-order rogue waves exhibits more richer structures, most of which have no counterparts in the corresponding local nonlinear equations.

Motivation & Objective

• To investigate the existence and dynamics of rogue waves in reverse time nonlocal nonlinear equations, which are fundamentally different from local equations due to time-reversal symmetry.
• To extend the Darboux transformation method to nonlocal integrable systems, particularly for the nonlocal DS hierarchy, to construct exact rogue wave solutions.
• To explore how free parameters in the solutions influence the wave dynamics, including boundedness and finite-time singularities.
• To compare the structural complexity of multi- and higher-order rogue waves in nonlocal systems with those in local equations, identifying new dynamical patterns.
• To establish a unified solution formula for both nonlocal DSI and DSII equations using a novel binary Darboux transformation.

Proposed method

• Employ the Darboux transformation (DT) method to generate exact rational solutions for the reverse time nonlocal NLS and DS equations.
• Introduce a unified binary Darboux transformation for the nonlocal DS system, enabling a single expression for solutions of both DSI and DSII equations.
• Apply symmetry reductions to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy to derive the reverse time nonlocal NLS and DS equations.
• Use the bilinear method and parameter-dependent solution formulas to analyze the behavior of fundamental and higher-order rogue waves.
• Numerically simulate and visualize solution dynamics under different parameter choices to classify bounded and singular behaviors.
• Verify solution validity by checking that the denominator functions remain real or non-vanishing under specific parameter conditions (e.g., purely imaginary f₁).

Experimental results

Research questions

• RQ1 Can rogue wave solutions exist in reverse time nonlocal nonlinear equations, where the solution at time t is coupled to the solution at time -t?
• RQ2 How do free parameters in the solution influence the dynamics of rogue waves—specifically, do they lead to bounded or finite-time singular solutions?
• RQ3 What novel dynamical patterns emerge in multi- and higher-order rogue waves in nonlocal systems compared to their local counterparts?
• RQ4 Can a unified Darboux transformation be constructed for both nonlocal DSI and DSII equations, and does it yield a single solution formula?
• RQ5 Do (1+2)-dimensional rogue waves in nonlocal DS equations exhibit hybrid behaviors such as localized lumps reflecting off parabolic rogue waves?

Key findings

• Fundamental (1+1)-dimensional rogue waves in the nonlocal NLS equation can be either bounded for all x and t or develop collapsing singularities, depending on the value of free parameters.
• Fundamental (1+2)-dimensional line rogue waves in the nonlocal DS equations can be bounded across all space and time or exhibit finite-time blowing-ups, again determined by parameter choices.
• Higher-order rogue waves in the nonlocal NLS equation display hybrid structures with triangular, pentagonal, and circular arrangements of collapsing and non-collapsing peaks, not seen in local equations.
• In the nonlocal DSI equation, second-order solutions show a unique dynamic pattern: a localized lump approaches from infinity, reflects off a parabolic-shaped rogue wave, and retreats back, with the parabola emerging from the background at t=0.
• For the nonlocal DSII equation, higher-order solutions with γ² = -1 yield only solutions with almost full-time singularities, and the existence of nonsingular or finite-time blowing-up solutions remains unknown.
• The unified binary Darboux transformation successfully generates solutions for both nonlocal DSI and DSII equations in a single, uniform expression, enabling systematic analysis of their dynamics.