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solution of triangle
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     solution;
     溶液;
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     T's solution.
     T解集中不出现空集;
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     b) The solution of the inequality.
     (2)解不等式;
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     Triangle Morphing
     三角形渐变动画
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Some explicit and exact travelling wave solutions for a class of the nonlinear Schrǒdinger equation involing quintic terms are obtained by two different ansatze.These solutions include solitary wave solutions of two types,singular travelling wave solutions and periodic wave solutions of triangle function type.

用两种不同的假设求出了一类带五次项的非线性Schrǒdinger 方程的显式精确行波解,这些解包括两种类型的孤波解、奇异行波解和三角函数型周期波解.

Precise soluability to a class of generalized symmetric regularized long wave equations with five order stronger nonlinear term are considered. Firstly, the problem of seek to solitary wave solutions of the equation are reduced to a problem to solve lienard equation with five order stronger nonlinear term u″(ξ)+lu(ξ)+mu 3(ξ)+nu 5(ξ) =0. Then, a equation 2 φ(ξ)φ″(ξ)-φ′ 2(ξ)+4lφ 2(ξ)+4mφ 3(ξ)+4nφ 4(ξ) =0 is obtained by a transformation u(ξ)=φ(ξ) . Finally, precise and explicit solutions...

Precise soluability to a class of generalized symmetric regularized long wave equations with five order stronger nonlinear term are considered. Firstly, the problem of seek to solitary wave solutions of the equation are reduced to a problem to solve lienard equation with five order stronger nonlinear term u″(ξ)+lu(ξ)+mu 3(ξ)+nu 5(ξ) =0. Then, a equation 2 φ(ξ)φ″(ξ)-φ′ 2(ξ)+4lφ 2(ξ)+4mφ 3(ξ)+4nφ 4(ξ) =0 is obtained by a transformation u(ξ)=φ(ξ) . Finally, precise and explicit solutions of two types for five order lienard equation are obtained by two different assumptions φ(ξ)=A e α(ξ+ξ 0) /[(1+ e α(ξ+ξ 0) ) 2+B e α(ξ+ξ 0) ] and φ(ξ)=A e α(ξ+ξ 0) /(1+ e α(ξ+ξ 0) ). According to this result, bell-shaped and kink-shaped solitary wave solutions for generalized symmetric regularized long wave equation are obtained, furthermore, singular travelling wave solutions of two types and periodic wave solutions of triangle function type are also given.

考虑了一类具5 次强非线性项的广义对称正则长波方程的精确可解性问题。首先将求此方程孤立波解的问题归结为求解具5 次强非线性项的Lienard 方程u″(ξ) + lu( ξ) + m u3(ξ) + nu5(ξ) =0 ,接着通过变换u(ξ) = φ(ξ) 得到φ( ξ) 满足的方程2 φ(ξ) φ″( ξ) - φ′2(ξ) + 4lφ2( ξ) + 4 mφ3(ξ) +4 nφ4(ξ) = 0 ,最后通过两种假设φ(ξ) = Aeα(ξ+ ξ0)/[(1 + eα(ξ+ ξ0))2 + Beα(ξ+ ξ0)] 和φ(ξ) = Aeα( ξ+ ξ0)/(1 +eα(ξ+ ξ0)) 获得了5 次Lienard 方程的二类显式精确解。据此求出了具5 次非线性项的广义对称正则长波方程的钟状和扭状孤立波解,并且获得此方程的两类奇异行波解和三角函数型周期波解

The modified unstable nonlinear Schro¨dinger equation is first reduced to a system of second order nonlinear partial differential equations,then some of the nonlinear algebraic equations and nonlinear ordinary differential equations are obtained by choosing different parameters. Some explicit and exact solutions of nonlinear ordinary differential equations,which are reduced above by a kind of combination of a direct method and ansatze method ,are found.And thus the explicit and exact solutions for...

The modified unstable nonlinear Schro¨dinger equation is first reduced to a system of second order nonlinear partial differential equations,then some of the nonlinear algebraic equations and nonlinear ordinary differential equations are obtained by choosing different parameters. Some explicit and exact solutions of nonlinear ordinary differential equations,which are reduced above by a kind of combination of a direct method and ansatze method ,are found.And thus the explicit and exact solutions for the modified unstable nonlinear Schro¨dinger equation are obtained.These solutions include the exact plane wave solutions,solitary wave solutions of the bell shape,solitary wave solutions of the kink shape,the singular travelling wave solutions,and periodic wave solutions of triangle function.

首先利用一个标准变换将修正的非稳非线性 Schro¨ dinger方程化成一个非线性偏微分方程组 ,接着通过选取不同参数得到一些非线性代数方程和非线性常微分方程。然后通过直接方法和假设方法的结合求得约化得到的非线性常微分方程的精确解 ,从而得到修正的非稳非线性 Schro¨ dinger方程的显式精确解 ,包括精确平面波解、钟状孤立波解、扭状孤立波解、奇异行波解和三角函数状周期波解

 
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