广义Camassa-Holm方程和修正Fornberg-Whitham方程的行波解及分支
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摘要
本文研究了结构相似的两个著名的数学物理方程的精确行波解：第一个方程是广义Camassa-Holm方程ut+2kux-uxxt+3u2ux=2uxuxx+uuxxx,(1)第二个方程是修正Fornberg-Whitham方程ut-uxxt+ux+u2ux=3uxuxx+uuxxx.(2)
利用微分方程动力系统分支方法、辅助方程法及数值模拟,对上述两个方程的精确行波解及解的分支进行了研究.
对方程(1)获得了以下结果：
(1°)我们证明k=8/3是几种类型的显式解的分支参数值.
(i)当k<8/3时,有下列五种形式的显式解：
(1)双曲型尖波解,
(2)分数形式的尖波解,
(3)分数形式的奇异波解,
(4)双曲型奇异波解,
(5)双曲型光滑孤立波解；
(ii)当k=8/3时,有两种显式解：分数形式的尖波解和分数形式的奇异波解；
(iii)当k>8/3时,没有显式非线性波解.
(2°)找到了某些临界波速值(也称分支波速值),使得孤立尖波(peakon)和反孤立尖波(anti-peakon)交替出现.
(3°)求出了双曲型尖波解和分数形式尖波解的临界波速值以及双曲型奇异波解和分数形式奇异波解的临界波速值.
对方程(2)获得了以下结果：
(1°)利用辅助方程法得到了其尖波解、孤立波解、三角函数解、椭圆函数解、爆破解这五类解的显式表达式,其中后三类解都是新的,而第二、三、四类解的表达式对任意的波速都成立,推广了前人的结果(He等只对特定的波速给出了孤立波和孤立尖波的显示表达式).
(2°)利用数学软件Mathematica验证了解的正确性,即将所得的解代入方程中,用计算机证明它们确为方程的解.在文中给出了验证程序和验证结果以及解的数值模拟结果(图形).
In this paper, we study the exact traveling wave solutions for two types of famous nonlinear equations. The first one is the generalized Camassa-Holm equation ut+2kux-uxxt+3u2ux=2uxuxx+uuxxx.(1)
The second is the modified Fornberg-Whitham equation ut-uxxt+ux+u2ux=3uxuxx+uuxxx.(2)
For Eq.(1) we obtain the following results:
(1°) It is verified that k=3/8is a bifurcation parametric value for several types of explicit nonlinear wave solutions.
(i) When k<3/8, there are five types of the explicit nonlinear wave solutions, which are
(1) hyperbolic peakon wave solution,
(2) fractional peakon wave solution,
(3) fractional singular wave solution,
(4) hyperbolic singular wave solution,
(5) hyperbolic smooth solitary wave solution.
(ⅱ) When k=3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.
(ⅲ) When k>3/8, there is not any type of explicit nonlinear wave solutions.
(2°) It is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.
(3°) It is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.
For Eq.(2) we get the following results:
(1°) Some explicit expressions of solutions are obtained by using auxiliary equation method, which include peakon wave solution, solitary wave solution, trigonometric func-tion solutions, elliptic function periodic solutions, and fractal type blow-up solutions. The latter three types of solution are new and the expressions of the second、the third and the forth types of solution are valid for any wave velocity c, which extend the results by other authors (He et. al just gave the explicit expressions of solitary wave solution for special wave velocity).
(2°) We use the software Mathematica to test the correctness of these solutions. That is, it is confirmed that these functions indeed satisfy Eq.(2) by computer. The testing programs and the testing results are given, and numerical simulation is also shown.
利用微分方程动力系统分支方法、辅助方程法及数值模拟,对上述两个方程的精确行波解及解的分支进行了研究.
对方程(1)获得了以下结果：
(1°)我们证明k=8/3是几种类型的显式解的分支参数值.
(i)当k<8/3时,有下列五种形式的显式解：
(1)双曲型尖波解,
(2)分数形式的尖波解,
(3)分数形式的奇异波解,
(4)双曲型奇异波解,
(5)双曲型光滑孤立波解；
(ii)当k=8/3时,有两种显式解：分数形式的尖波解和分数形式的奇异波解；
(iii)当k>8/3时,没有显式非线性波解.
(2°)找到了某些临界波速值(也称分支波速值),使得孤立尖波(peakon)和反孤立尖波(anti-peakon)交替出现.
(3°)求出了双曲型尖波解和分数形式尖波解的临界波速值以及双曲型奇异波解和分数形式奇异波解的临界波速值.
对方程(2)获得了以下结果：
(1°)利用辅助方程法得到了其尖波解、孤立波解、三角函数解、椭圆函数解、爆破解这五类解的显式表达式,其中后三类解都是新的,而第二、三、四类解的表达式对任意的波速都成立,推广了前人的结果(He等只对特定的波速给出了孤立波和孤立尖波的显示表达式).
(2°)利用数学软件Mathematica验证了解的正确性,即将所得的解代入方程中,用计算机证明它们确为方程的解.在文中给出了验证程序和验证结果以及解的数值模拟结果(图形).
In this paper, we study the exact traveling wave solutions for two types of famous nonlinear equations. The first one is the generalized Camassa-Holm equation ut+2kux-uxxt+3u2ux=2uxuxx+uuxxx.(1)
The second is the modified Fornberg-Whitham equation ut-uxxt+ux+u2ux=3uxuxx+uuxxx.(2)
For Eq.(1) we obtain the following results:
(1°) It is verified that k=3/8is a bifurcation parametric value for several types of explicit nonlinear wave solutions.
(i) When k<3/8, there are five types of the explicit nonlinear wave solutions, which are
(1) hyperbolic peakon wave solution,
(2) fractional peakon wave solution,
(3) fractional singular wave solution,
(4) hyperbolic singular wave solution,
(5) hyperbolic smooth solitary wave solution.
(ⅱ) When k=3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.
(ⅲ) When k>3/8, there is not any type of explicit nonlinear wave solutions.
(2°) It is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.
(3°) It is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.
For Eq.(2) we get the following results:
(1°) Some explicit expressions of solutions are obtained by using auxiliary equation method, which include peakon wave solution, solitary wave solution, trigonometric func-tion solutions, elliptic function periodic solutions, and fractal type blow-up solutions. The latter three types of solution are new and the expressions of the second、the third and the forth types of solution are valid for any wave velocity c, which extend the results by other authors (He et. al just gave the explicit expressions of solitary wave solution for special wave velocity).
(2°) We use the software Mathematica to test the correctness of these solutions. That is, it is confirmed that these functions indeed satisfy Eq.(2) by computer. The testing programs and the testing results are given, and numerical simulation is also shown.
引文
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