Source-linked AI summary
Rogue wave and a pair of resonance stripe solitons to a reduced generalized (3+1)-dimensional KP equation
Xiaoen Zhang, Yong Chen, Xiaoyan Tang
TL;DR
The paper studies lump solutions and their interactions for a reduced (3+1)-dimensional KP equation. Using bilinear and symbol methods, it finds stripe-soliton swallowing and a rogue wave generated by resonance-soliton interaction.
Problem
The paper investigates lump solutions and their interaction with stripe solitons in a generalized (3+1)-dimensional KP equation.
Method
The authors use Hirota bilinear formulation, symbol calculation, positive quadratic functions, and hyperbolic or exponential functions to construct and analyze solutions.
Results
The lump soliton is swallowed by one stripe soliton, while interaction with a pair of resonance stripe solitons produces a short-lived, high-amplitude rogue wave.
Takeaways & Limitations
The constructed solutions exhibit rational localization, soliton absorption, and rogue-wave generation through resonance-soliton interaction.
Takeaways & Limitations
Some parameter choices do not produce lump solitons, so the construction requires nonzero conditions and parameter constraints.
Abstract
from arXiv · showhide
Based on the bilinear operator and symbol calculation, some lump solutions are presented, rationally localized in all directions in the space, to a reduced (3+1)-dimensional KP equation. The lump solutions all contain six parameters, four of which must cater to the non-zero conditions so as to insure the analyticity and rational localization, while the others are free. Then the interaction between lump soliton and one stripe soliton is described and the result shows that the lump soliton will be drowned or swallowed by the stripe soliton. Furthermore, we extend this method to a new combination of positive quadratic function and hyperbolic functions. Especially, it is interesting that a rogue wave is found to be aroused by the interaction between lump soliton and a pair of resonance stripe solitons. By choosing the values of the parameters, the dynamic properties of lump solution, interaction between lump soliton and one stripe soliton, rogue wave, generated by the interaction between lump soliton and a pair of resonance solitons, are shown graphically.
1. Introduction
The introduction situates lump and rogue-wave solutions within integrable-soliton research and states the paper’s focus on their interactions in a reduced (3+1)-dimensional KP equation.
- The Hirota bilinear method is highlighted as a popular approach for studying solutions of integrable nonlinear equations.
- Lump solutions are rationally localized in all spatial directions, unlike the rogue-wave solutions discussed as another special rational-solution type.
- The paper studies lump solutions and the interaction between a lump soliton and one stripe soliton in a generalized (3+1)-dimensional KP equation.
- The method is extended from a positive quadratic function to a combination with hyperbolic cosine, producing a rogue wave through interaction with a pair of resonance solitons.
2. Lump solution of a reduced generalized (3+1)-dimensional KP equation
A positive quadratic-function ansatz and the transformation u = 4(ln f)_xx generate lump solutions whose parameter constraints ensure analyticity and rational localization.
- The solution is obtained through the transformation u = 4(ln f)_xx after introducing a bilinear formulation.
- The ansatz uses f = g^2 + h^2 + a9, with g and h linear in x, y, and t and parameters a_i to be determined.
- Parameter conditions are imposed to keep f analytical and positive and to guarantee rational localization of u in all directions in the (x, y)-plane.
- The resulting lump solutions are visualized with profiles at y = 0 and t = 0 and with density plots showing their shape.
3. The interaction between lump soliton and one stripe soliton
The paper combines a positive quadratic function with an exponential function to model lump–stripe interaction, finding that the lump is gradually swallowed by the stripe soliton.
- The interaction ansatz combines a positive quadratic function with one exponential function.
- The construction uses linear forms m1 and n1 together with an exponential phase l1 in the combined solution.
- Parameter conditions ensure that the corresponding solution is positive, analytical, and localized in all directions in the (x, y)-plane.
- As time advances, the lump’s energy transfers gradually into the stripe soliton until the lump is completely swallowed and the two structures continue as one soliton.
4. Rogue wave and a pair of resonance solitons
The paper represents two resonance solitons through exponential or hyperbolic-cosine terms and analyzes their interaction with a lump soliton. The evolution produces a transient rogue wave between the resonance stripes.
- The two-stripe construction combines a positive quadratic function with two exponential functions to seek a bilinear-form solution.
- The two exponential functions are identified as a pair of resonance solitons and transformed into a hyperbolic cosine representation.
- The parameter relations are selected so the resulting solution remains positive, analytical, and localized in all directions in the (x, y)-plane.
- As t approaches ±∞, only the pair of resonance solitons remains visibly present, while the lump becomes more apparent at finite times.
- At t = 0, a rogue wave derived from the lump appears between the resonance solitons, links them, and then transfers toward the other stripe soliton.
5. Discussion
The discussion establishes parameter and determinant constraints for analytic, localized lump solutions, while showing distinct interaction outcomes with stripe solitons. It also identifies a transient rogue wave generated by lump–resonance-stripe interaction and notes broader extensions for future study.
- A non-zero determinant and parameter constraints guarantee analyticity and rational localization of the obtained lump solution.
- When a9 < 0, the quadratic function cannot support a lump soliton, so collision analysis is unavailable for that case.
- The lump soliton initially maintains its type and energy spread but is ultimately swallowed by the stripe soliton after interaction.
- A rogue wave emerges when a lump soliton interacts with a pair of resonance stripe solitons, reaches a short-lived maximum amplitude, and later degenerates into a lump soliton.
- The authors propose extending the interaction analysis to other soliton types and discrete equations.