arXiv 1601.063302016-01-24nlin.PS

大量实非线性方程新的PT不变扭解和脉冲解

Novel PT-invariant Kink and Pulse Solutions For a Large Number of Real Nonlinear Equations

Avinash Khare, Avadh Saxena

备注：: 38 pages, no figures. arXiv admin note: text overlap with arXiv:1509.02899

摘要：

对于大量的实非线性方程，无论是连续的或离散的，可积的或不可积的，非耦合的或耦合的，我们表明，只要一个真正的非线性方程承认扭结解的$\tanh βx$，其中$β$是逆的扭结宽度，它也承认解决方案的PT不变的组合$\tanh 2βx \pm i \sech 2 βx$，即扭结宽度减少了一半的真正的扭结解。我们表明，扭结和PT不变扭结是线性稳定的，并获得表达式的零模式的情况下，几个PT不变扭结解决方案。进一步，对于一些实非线性方程，我们证明了只要一个非线性方程在$\sn(x,m)$中存在周期扭折解，它在PT不变组合$\sn(x,m) \pm i \cn(x,m)$和$\sn(x,m)\pm i \dn(x,m)$中也存在周期解.最后，对于耦合方程，我们证明了一个人不仅可以有复杂的PT不变的解决方案，PT特征值$+1$或$-1$在这两个领域，但也可以有解决方案，PT特征值$+1$在一个领域和$-1$在其他领域。

英文摘要：

For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, uncoupled or coupled, we show that whenever a real nonlinear equation admits kink solutions in terms of $\tanh βx$, where $β$ is the inverse of the kink width, it also admits solutions in terms of the PT-invariant combinations $\tanh 2βx \pm i \sech 2 βx$, i.e. the kink width is reduced by half to that of the real kink solution. We show that both the kink and the PT-invariant kink are linearly stable and obtain expressions for the zero mode in the case of several PT-invariant kink solutions. Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits periodic kink solutions in terms of $\sn(x,m)$, it also admits periodic solutions in terms of the PT-invariant combinations $\sn(x,m) \pm i \cn(x,m)$ as well as $\sn(x,m)\pm i \dn(x,m)$. Finally, for coupled equations we show that one cannot only have complex PT-invariant solutions with PT eigenvalue $+1$ or $-1$ in both the fields but one can also have solutions with PT eigenvalue $+1$ in one field and $-1$ in the other field.

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