鄭曉翠,高曉紅
(1.西北大學(xué)數(shù)學(xué)學(xué)院,陜西西安710127;2.西北大學(xué)非線性科學(xué)研究中心,陜西西安710069)
一類三階非線性色散波方程Cauchy問(wèn)題解的解析性
鄭曉翠1,高曉紅2
(1.西北大學(xué)數(shù)學(xué)學(xué)院,陜西西安710127;2.西北大學(xué)非線性科學(xué)研究中心,陜西西安710069)
利用抽象的Cauchy-Kowalevski定理,證明了一類三階非線性色散方程Cauchy問(wèn)題解的解析性,即如果該Cauchy問(wèn)題初值是解析的,則其解關(guān)于空間變量是全局解析的,關(guān)于時(shí)間變量是局部解析的.
非線性色散波方程;Cauchy-Kowalevski定理;解析性
研究了一類三階非線性色散方程,方程形式如下:
這里g[u]=κu+αu2+βu3,其中α,β,κ,θ,λ均是常參數(shù).
方程(1)包含許多有意義的物理模型,其可以描述連續(xù)介質(zhì).例如:當(dāng)
且θ為任意實(shí)參數(shù)時(shí),方程(1)就可以作為一個(gè)新的模型來(lái)描述圓柱形可壓縮超彈性桿[1-2].其中,各種可壓縮材料的物理參數(shù)θ的取值范圍是從-29.4760到3.4174.此外,當(dāng)參數(shù)滿足某些條件時(shí),方程(1)可以約化成一些淺水波方程.文獻(xiàn)[3]中討論了在具體參數(shù)條件下,方程(1)的對(duì)稱性.文獻(xiàn)[4]中又給出了該模型是完全可積的.
在此研究基礎(chǔ)上,將考慮方程(1)的Cauchy問(wèn)題:解的解析性.
本文的主要方法是在一個(gè)合適度量的Banach空間中利用抽象的Cauchy-Kowalevski定理來(lái)證明.這個(gè)方法是由文獻(xiàn)[5]引入的,之后在文獻(xiàn)[6-7]中得到更一般的結(jié)論.
在這一節(jié),將引入在證明過(guò)程中所需要的一些基本知識(shí)和引理.
首先給出結(jié)論.
參考文獻(xiàn)
[1]Dai H H.Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[J].Acta Mech.,1998,127:193-207.
[2]Dai H H,Huo Y.Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[J].Proc.R.Soc.,2000,456:331-363.
[3]Clarkson P A,Mansfield E L,Priestley T J.Symmetries of a class of nonlinear third order partial differential equations[J].Math.Comput.,1997,25:195-212.
[4]Ivanov R.On the integrability of a class of nonlinear dispersive wave equation[J].J.Nonlinear Math.Phys.,2005,12(4):462-468.
[5]Ovsiannikov L V.A nonlinear cauchy problem in a scale of Banach spaces[J].Dokl.Akad.Nauk.SSSR.,1971,200:789-792.
[6]Himonas A A,Misiolek G.Analyticity of the Cauchy problem for an integrable evolution equation[J].Math. Ann.,2003,327:575-584.
[7]Yan K,Yin Z Z.Analytic solutions of the Cauchy problem for two-component shallow water systems[J]. Math.Z.,2011,269:1113-1127.
[8]Baouendi S,Goulaouic C.Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems[J]. Comm.Partial.Diff.Eqs.,1977,2:1151-1162.
Analyticity of the Cauchy problem for a class of third-order dispersive wave equations
Zheng Xiaocui1,Gao Xiaohong2
(1.College of Mathematics,Northwest University,Xi′an710127,China 2.Center for Nonlinear Studies,Northwest University,Xi′an 710069,China)
Using the abstract Cauchy-Kowalevski theorem,the analytic solutions of the Cauchy problems for a class of third-order nonlinear dispersive wave equations are discussed.It is proved that if the initial values of this Cauchy problems are analytic,then their solutions are analytic in both variables,globally in space and locally in time.
nonlinear dispersive wave equations,Cauchy-Kowalevski theorem,analyticity
O175.29
A
1008-5513(2016)02-0190-07
10.3969/j.issn.1008-5513.2016.02.010
2015-09-28.
國(guó)家自然科學(xué)基金面上項(xiàng)目(11471259);陜西省自然科學(xué)基礎(chǔ)研究計(jì)劃青年項(xiàng)目(2014JQ1002).
鄭曉翠(1988-),碩士生,研究方向:偏微分方程.
2010 MSC:35B30
純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué)2016年2期