AI 中文总结
本文研究结合Camassa-Holm方程非线性结构与Zakharov-Kuznetsov方程横向色散的新型二维CH-ZK方程，建立其局部适定性与爆破准则，构造有限时间爆破强解，证明解的唯一延拓性质，且得到行波解的刚性定理。
AI 中文摘要
本文研究一种新型二维非线性色散波模型，即Camassa-Holm-Zakharov-Kuznetsov（CH-ZK）方程，该方程结合了Camassa-Holm方程的非线性结构与Zakharov-Kuznetsov方程的横向拉普拉斯色散特性。我们首先在合适的Sobolev空间中建立其柯西问题的局部适定性，并推导强解的爆破准则；随后构造了CH-ZK方程的有限时间爆破强解；此外，证明了CH-ZK方程解的唯一延拓性质；最后，研究了尖峰孤立波与光滑孤立波的存在性，并根据波速大小得到了行波解的刚性定理。
英文摘要
This paper is devoted to a new two-dimensional nonlinear dispersive wave model named as the Camassa-Holm-Zakharov-Kuznetsov (CH-ZK) equation which combines the nonlinear structure of the Camassa-Holm equation with the transverse Laplacian dispersion of the Zakharov-Kuznetsov equation. We first establish the local well-posedness of its Cauchy problem in a suitable Sobolev space and derive a blow-up criterion for strong solutions. Then the finite-time blow-up strong solutions for the CH-ZK equation have been constructed. Moreover, we prove a unique continuation property for the solutions to the CH-ZK equation. Finally, we investigate the existence of both peaked and smooth solitary waves, and obtain a rigidity theorem for the traveling wave solutions according to the magnitude of wave speed.
Comments24 pages
