（２＋１）维非线性色散长波方程的相似约化和解析解

Abstract

Abstract:

首先借助于Ｍａｔｈｅｍａｔｉｃａ软件，将Ｃｌａｒｋｓｏｎ和Ｋｒｕｓｋａｌ引入的直接约化法推广并应用于（２＋１）维偏微分方程组情形（２＋１）维非线性色散长波方程，获得了该方程的六种类型的相似约化和若干解析解，其中包括ＰａｉｎｌｅｖｅⅡ型方程和孤子解．然后基于文［５］的结论，通过引入新的级数变换，获得了该方程的有理分式解析解．这种方法也适合于其它的微分方程．

Key words: 非线性波动方程, 直接约化法, 相似约化, 解析解, ＰａｉｌｅｖｅⅡ型方程

CLC Number:

YAN Zhen-Ya, ZHANG Hong-Qiang. Similarity Reductions and Analytic Solutions for (2+1)-dimensional Dispersive Long Wave Equation[J].Acta mathematica scientia,Series A, 2001, 21(3): 384-390.

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１　ＡｂｌｏｗｉｔｚＭＪ，ＣｌａｒｋｓｏｎＰＡ．Ｓｏｌｉｔｏｎｓ，Ｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓａｎｄｉｎｖｅｒｓｅｓｃａｔｔｅｒｉｎｇ．Ｃａｍｂｒｉｄｇｅ：Ｃａｍｂｒｉｄｇｅ
ＵｎｉｖｅｒｓｉｔｙＰｒｅｓｓ，１９９１
２　ＧｕＣＨ，ｅｔａｌ．Ｓｏｌｉｔｏｎｔｈｅｏｒｙａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｓ．Ｂｅｒｌｉｎ：Ｓｐｒｉｎｇｅｒ，１９９５
３　ＷｉｔｈａｍＧＢ．Ｖａｒｉａｔｉｏｎａｌｍｅｔｈｏｄｓａｎｄａｐｐｌｉｃａｔｉｏｎｓｔｏｗａｔｅｒｗａｖｅ．ＰｒｏｃＲｏｙＳｏｃＬｏｎｄｏｎ，ＳｅｒＡ．１９６７，２９９：６－２５
４　ＫａｕｐＤＪ．Ａｈｉｇｈｅｒｏｒｄｅｒｗａｔｅｒｗａｖｅｅｑｕａｔｉｏｎａｎｄｍｅｔｈｏｄｆｏｒｓｏｌｖｉｎｇｉｔ．ＰｒｏｃＴｈｅｏｒＰｈｙｓ，１９７５，５４：３９６－４０８
５　ＷａｎｇＭＬ，ｅｔａｌ．Ａｐｐｌｉｃａｔｉｏｎｏｆａｈｏｍｏｇｅｎｅｏｕｓｂａｌａｎｃｅｍｅｔｈｏｄｔｏｅｘａｃｔｓｏｌｕｔｉｏｎｓｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓｉｎｍａｔｈｅ
ｍａｔｉｃａｌｐｈｙｓｉｃｓ．ＰｈｙｓＬｅｔｔＡ，１９９６，２１６：６７－７５
６　ＣｌａｒｋｓｏｎＰＡ，ＫｒｕｓｋａｌＭＤ．ＮｅｗｓｉｍｉｌａｒｉｔｙｒｅｄｕｃｔｉｏｎｓｏｆＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎ．ＪＭａｔｈＰｈｙｓ，１９８９，３０：２２０１－２２１３
７　ＣｌａｒｋｓｏｎＰＡ，ｅｔａｌ．ＳｙｍｍｅｔｒｙｒｅｄｕｃｔｉｏｎｓｏｆａｇｅｎｅｒａｌｉｚｅｄｃｙｌｉｎｄｒｉｃａｌＳｃｈｒｏｇｉｎｇｅｒｅｑｕａｔｉｏｎ．ＪＰｈｙｓＡ，１９９３，２６：１３３
－１５０
８　ＬｏｕＳＹ．ＳｉｍｉｌａｒｉｔｙｓｏｌｕｔｉｏｎｓｏｆｔｈｅＫＰｅｑｕａｔｉｏｎ．ＪＰｈｙｓＡ，１９９０，２３：６４９－６５４
９　闫振亚等．非线性ＭＭＷＤ方程组的对称约化和显式精确解．力学与实践，１９９９，２０（３）：４８－５０
１０　闫振亚等．非线性浅水长波方程组的显式精确解．物理学报，１９９９，４８（１１）：１９６２－１９６８

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Similarity Reductions and Analytic Solutions for (2+1)-dimensional Dispersive Long Wave Equation

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