广义21维kp-bbm方程和广义camassa-holm方程的行波解-traveling wave solutions of generalized 21 - dimensional kp - bbm equation and generalized ca massa - holm equation.docx

摘要本文研究了两个非线性微分方程：广义（2+1）维KP-BBM方程和广义Camassa-Holm（GCH）方程．利用sine-cosine方法、扩展tanh方法获得了广义（2+1）维KP-BBM方程的紧解、孤子类解和周期解．运用动力系统分支理论方法以及直接积分法得到了广义Camassa-Holm（GCH）方程在????2??情形下的peakon解、cuspon解、光滑孤子解和周期尖波解；在????3??情况下的peakon解、圈解(loopsolution)和周期尖波解；在????0情况下的无界波解，周期波解.同时我们给出了上述解存在的各类充分条件，并讨论了Camassa-Holm方程的解的动力学行为．本文的结构安排如下：第一章，主要阐述了广义（2+1）维KP-BBM方程和广义Camassa-Holm（GCH）方程的研究背景及研究现状．第二章，介绍了孤子理论概况及本文运用的方法：sine-cosine方法、扩展tanh方法，动力系统分支理论方法．第三章，讨论了广义（2+1）维KP-BBM方程的紧孤子解、孤子类解和周期解．第四章，研究了广义Camassa-Holm（GCH）方程在????2??,情况下的行波解并对解进行了分类．第五章，对本文进行了总结，并提出对未来工作的展望．????3??,????0关键词：广义（2+1）维KP-BBM方程；广义Camassa-Holm方程；动力系统分支理论；单峰孤子；孤立波IAbstractInthispaperwestudytwononlineardifferentialequations：(2+1)-dimensionalgeneralizedKP-BBMequationandgeneralizedCamassa-Holmequation.Byusingthesine-cosinemethodandtheextendedtanh-method,weobtaincompactonsolution,solitarypatternsolutionandperiodicsolutionofthe(2+1)-dimensionalgeneralizedKP-BBMequation.ForgeneralizedCamassa-Holmequation,byusingthemethodofbifurcationtheoryofdynamicalsystemanddirectintegrationmethod,weobtainpeakon,cuspon,smoothsoliton,periodiccuspwavesolutionsinthecase????2??,obtainpeakon,loop,periodiccuspwavesolutionsinthecase????3??,andobtainunboundedwave,periodicwavesolutionsinthecase????0．Inaddition,wegivevarioussufficientconditionstoguaranteetheexistenceoftheabovesolutionsanddiscussthetravelingwavesolutionbranchanddynamicbehaviorofthisequation．Thestructureofthepaperisorganizedasfollows：Thefirstchapter,wemainlytalkabouttheresearchbackgroundandresearchdevelopmentsofthe(2+1)-dimensionalgeneralizedKP-BBMequationandgeneralizedCamassa-Holmequation．Thesecondchapter,weintroducetheoverviewofsolitontheoriesandthesine-cosinemethod,theextendedtanh-methodmethod,themethodofbifurcationtheoryofdynamicalsystemwhichareappliedtothispaper．Thethirdchapter,wediscussthecompactonsolution,solitarypatternsolutionandperiodicsolutionofthe(2+1)-dimensionalgeneralizedKP-BBMequation．Thefourthchapter,westudythetravelingsolutionsandsolutionclassificationsofgeneralizedCamassa