无色散BKP方程族可积耦合推广及其求解

数学物理学报 ›› 2017, Vol. 37 ›› Issue (2): 313-325.

无色散BKP方程族可积耦合推广及其求解

刘晶鑫¹, 吴红霞¹, 曾云波²

1 集美大学理学院数学系福建厦门 361021;
2 清华大学数学科学系北京 100084

收稿日期:2016-01-14 修回日期:2016-10-24 出版日期:2017-04-26 发布日期:2017-04-26
通讯作者: 吴红霞 E-mail:wuhongxia@jmu.edu.cn
基金资助:
国家自然科学基金（11201178）、福建省出国留学奖学金和集美大学科研启动基金

On the Ingegrable Coupled Generalization of Dispersionless BKP Hierarchy and Its Solutions

Liu Jingxin¹, Wu Hongxia¹, Zeng Yunbo²

1 Department of Mathematics, Jimei University, Fujian Xiamen 361021;
2 Department of Mathematical, Tsinghua University, Beijing 100084

Received:2016-01-14 Revised:2016-10-24 Online:2017-04-26 Published:2017-04-26
Supported by:
Supported by the NSFC (11201178), the Fujian Provincial Visiting Scholar Program and the Scientific Research Foundation of Jimei University

摘要/Abstract

摘要： 该文通过对B类Kadomtsev-Petviashvili（B type of Kadomtsev-Petviashvili，简称为BKP）方程族基于特征函数及共轭特征函数表示的对称约束取无色散极限，得到无色散BKP（dispersionless BKP，简称为dBKP）方程族的对称约束；其次，基于dBKP方程族的对称约束，考察了dBKP方程族的推广问题.通过计算推广的dBKP方程族的零曲率方程，该文导出了第一、二类型的带自相容源的dBKP方程（dispersionless BKP equation with self-consistent sources，简称为dBKPESCS）及其相应的守恒方程.最后，利用速端变换及约化的方法求解了第一型dBKPESCS.

关键词: 无色散BKP方程族, 推广, 对称约束, 带自相容源的无色散BKP方程族, 速端解

Abstract: The symmetry constraint for dispersionless BKP (dBKP) hierarchy is firstly derived by taking dispersionless limit of that for BKP hierarchy. Then, based on the symmetry constraint for dBKP hierarchy, a new extended dBKP hierarchy is constructed. In addition, the integrability of this new extended dBKP hierarchy is proved by presenting its zero-curvature equation and the related conservation equation. From its zero-curvature equation, two types of dBKP equations with self-consistent sources (dBKPESCS) together with their associated conservation equations are obtained. Hodograph solutions for the first type of dBKPESCS are finally obtained.

Key words: Dispersionless BKP hierarchy, Extended, Symmetry constraint, Dispersionless BKP hierarchy with self-consistent sources, Hodograph solutions

中图分类号:

刘晶鑫, 吴红霞, 曾云波. 无色散BKP方程族可积耦合推广及其求解[J]. 数学物理学报, 2017, 37(2): 313-325.

Liu Jingxin, Wu Hongxia, Zeng Yunbo. On the Ingegrable Coupled Generalization of Dispersionless BKP Hierarchy and Its Solutions[J]. Acta mathematica scientia,Series A, 2017, 37(2): 313-325.

参考文献

[1] Mineev-Weinstein M,Wiegmann P B,Zabrodin A.Integrable structure of interface dynamics.Phys Rev Lett,2000,84(22):5106-5109
[2] Gibbons J,Tsarev S P.Reductions of the Benney equations.Phys Lett A,1996,211(1):19-24
[3] Gibbons J,Tsarev S P.Conformal maps and reductions of the Benney equation.Phys Lett A,1999,258(4-6):263-271
[4] Kodama Y,Gibbons J.A method for solving the dispersionless KP equation and its exact solutions.Phys Lett A,1988,129:223-226
[5] Krichever I M.The dispersionless Lax equations and topological minimal models.Commun Math Phys,1992,143(2):415-429
[6] Aoyama S,Kodama Y.Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP equation.Commun Math Phys,1996,182(1):185-219
[7] Lax P D,Levermore C D.The zero dispersion limit for the KdV equation I,Ⅱ,Ⅲ.Commun Pure Appl Math,1983,36:253-290
[8] Takasaki T,Takebe T.Integrable hierarchies and dispersionless limit.Rev Math Phys,1995,47:743-808
[9] Takasaki T,Takebe T.Quasi-classical limit of KP hierarchy,W-symmetries and free fermions.J Math Sci,1999,94(4):1635-1641
[10] Konopelchenko B,Martinez Alonso L,Ragnisco O.The∂-approach to the dispersionless KP hierarchy.J Phys A,2001,34(47):10209-10217
[11] Tasakaki K.Quasi-classical limit of BKP hierarchy and W-infinity symmetries.Letters in Mathematical Physics,1993,28:177-185
[12] Dasgupta N,Roy A.A new integrable equation in the 2+1-dimension-dispersionless limit of the BKP equation.J Phys A:Math Gen,1992,25:1033-1038
[13] Chen Y T,Tu M H.A note on the dispersionless BKP hierarchy.J Math A:Math Gen,2006,39:7641-7655
[14] Bogdanov L V,Konopelchenko B G.Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type.Physics Letters A,2004,6:448-459
[15] VandeLeur J W.Schlesinger-Bäcklund transformation for N-component KP.J Math Phys,1998,39:2833-2847
[16] Kacand V G,VandeLeur J W.The N-component KP hierarchy and representation theory.J Math Phys,2003,44:3245-3293
[17] Mel'nikov V K.On equation for wave interaction.Lett Math Phys,1983,7:129-136
[18] Hu X B,Wang H Y.Construction of dKP and BKP equations with self-consistent sources.Inverse Probl,2006,22:1903-1920
[19] Xiao T,Zeng Y B.The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent soueces.J Nonlinear Math Phys,2006,13:193-204
[20] Liu X J,Zeng Y B,Lin R L.A new extended KP hierarchy.Physics Letters A,2008,372:3819-3823
[21] Xiao T,Zeng Y B.The constrained dispersionless mKP hierarchy and the dispersionless mKP hierarchy with self-consistent sources.Phys Lett A,2006,349:128-134
[22] Cheng J P,He J S.The integral type gauge transformation and the additional symmetry for the constrained KP hierarchy.Acta Mathematica Scientia,2015,35B(5):1111-1121
[23] Date E,Jimbo M,Kashiwara M,et al.Transformation groups for soliton equations//Nonlinear Integrable Systems-Classical Theory and Quantum Theory.Koyo Singapore:World Scientific,1983:39-119
[24] Loris I,Willox R.Symmetry reductions of the BKP hierarchy.J Math Phys,1999,40:1420-1431