论文标题
分数修改的Korteweg--de Vries方程中的周期波
Periodic waves in the fractional modified Korteweg--de Vries equation
论文作者
论文摘要
在分数laplacian的情况下，重新审视了修改后的Korteweg-De Vries（MKDV）方程中的周期性波。在当地情况下的两个溶液家族由标志性脱氧和标志性的cNOIDALITE溶液给出。两种溶液在一般的分数情况下都可以表征为能量功能的二次最小化部分，但可承担固定的$ l^4 $ norm：sign-definite（sign-definite）溶液，在偶数（奇数）函数的子空间中获得。对于解决方案，计算了Morse索引，并得出了光谱稳定性标准。我们从数字上表明，标志性解决方案的家族对较低的规律性的分数拉普拉雄雄性具有通用的折叠分叉，并且在分数和局部病例中，Sign-Indefinite溶液的家族都具有通用的对称性分叉。
Periodic waves in the modified Korteweg-de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is derived. We show numerically that the family of sign-definite solutions has a generic fold bifurcation for the fractional Laplacian of lower regularity and the family of sign-indefinite solutions has a generic symmetry-breaking bifurcation both in the fractional and local cases.
