In this work, we consider the initial-value problem associated with a coupled system of generalized Korteweg–de Vries equations. We present a relationship between the best constant for a Gagliardo–Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary wave solutions with minimal mass, the so-called ground state solutions. A characterization of the ground states and the orbital instability of the solitary waves are also established.

中文翻译：

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gKdV方程耦合系统的孤波解和整体适定性

在这项工作中，我们考虑与广义Korteweg-de Vries方程耦合系统相关的初值问题。我们提出了Gagliardo-Nirenberg型不等式的最佳常数与能量空间中整体解的存在准则之间的关系。我们证明了这样一个常数与质量最小的孤立波解的存在问题直接相关，即所谓的基态解。还建立了基态的表征和孤立波的轨道不稳定性。