{"title":"Davey-Stewartson II方程中一般高阶块解的预测","authors":"Xue-Wei Yan, Haie Long, Yong Chen","doi":"10.1098/rspa.2023.0455","DOIUrl":null,"url":null,"abstract":"Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive, the triangle lump reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction when time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Prediction of general high-order lump solutions in the Davey–Stewartson II equation\",\"authors\":\"Xue-Wei Yan, Haie Long, Yong Chen\",\"doi\":\"10.1098/rspa.2023.0455\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time <jats:inline-formula> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive, the triangle lump reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction when time <jats:inline-formula> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive. 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引用次数: 0
摘要
本文研究的是 Davey-Stewartson (DS) II 方程。基于 Kadomtsev-Petviashvili (KP) 简化法和舒尔多项式理论,我们构建了一般高阶块体解。通过提取真解的舒尔多项式的前阶渐近值,得出了由基本块体及其位置组成的预测解。当选择解的索引为不同的正整数组合时,大时间的预测解反映了真解的两类块状模式。第一类是三角形的块状模式,由 Yablonskii-Vorob'ev 多项式的根结构分析描述。当时间 t 由大负值变为大正值时,三角形凸块沿 y 方向反转。第二类块状模式包括外部区域的非三角形,它由弗伦斯基-赫米特多项式的非零根结构分析描述,以及内部区域的可能三角形,它由雅布隆斯基-沃罗布夫多项式的根结构分析描述。此外,当时间 t 从大负值变为大正值时,外部区域的非三角形块状图案会旋转一个角度,而内部区域的可能三角形块状图案则会沿 y 方向反转。这些结果加深了我们对高阶凸块时间演化机制的理解。
Prediction of general high-order lump solutions in the Davey–Stewartson II equation
Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time t evolves from large negative to large positive, the triangle lump reverses itself along the y -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the y -direction when time t evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.
期刊介绍:
Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.