We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for N − soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the degrees of modifications of the charges around the soliton interaction regions, we compute numerically some representative anomalies, associated to the lowest order quasi-conservation laws, depending on the deformation parameters { ϵ 1 , ϵ 2 }, which include the standard KdV ( ϵ 1 = ϵ 2 = 0), the regularized long-wave (RLW) ( ϵ 1 = 1 , ϵ 2 = 0), the modified regularized long-wave (mRLW) ( ϵ 1 = ϵ 2 = 1) and the KdV-RLW (KdV-BBM) type ( ϵ 2 = 0 , ≠ = {0, 1}) equations, respectively. Our numerical simulations show the elastic scattering of two and three solitons for a wide range of values of the set { ϵ 1 , ϵ 2 }, for a variety of amplitudes and relative velocities. The KdV-type equations are quite ubiquitous in several areas of non-linear science, and they find relevant applications in the study of General Relativity on AdS 3 , Bose-Einstein condensates, superconductivity and soliton gas and turbulence in fluid dynamics.

中文翻译：

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准可积 KdV 模型、无限数量异常电荷的塔和孤子碰撞

我们通过分析和数值方法发现了用于 Korteweg-de Vries 方程 (KdV) 变形的无限​​数量渐近守恒电荷的新塔。分析表明，标准 KdV 也表现出一些具有无限数量异常电荷的塔，并且它们的相关异常对于 N - 孤子解消失了。KdV 模型的一些变形是通过 Riccati 型赝势方法执行的，并且使用变形模型的线性公式提供了无数精确的非局部守恒定律。为了检查孤子相互作用区域周围电荷的修改程度，我们根据变形参数 { ϵ 1 , ϵ 2 }，数值计算了一些与最低阶准守恒定律相关的代表性异常，其中包括标准 KdV ( ϵ 1 = ϵ 2 = 0)、正则化长波 (RLW) ( ϵ 1 = 1 , ϵ 2 = 0)、修正的正则化长波 (mRLW) ( ϵ 1 = ϵ 2 = 1) 和 KdV-RLW (KdV-BBM) 类型 ( ϵ 2 = 0 , ≠ = {0, 1}) 方程，分别。我们的数值模拟显示了两个和三个孤子的弹性散射，适用于各种振幅和相对速度的集合 { ϵ 1 , ϵ 2 } 的广泛值。KdV 型方程在非线性科学的几个领域中非常普遍，它们在 AdS 3 的广义相对论、玻色-爱因斯坦凝聚、超导性和孤子气体以及流体动力学中的湍流的研究中找到了相关应用。分别是修正的正则化长波 (mRLW) ( ϵ 1 = ϵ 2 = 1) 和 KdV-RLW (KdV-BBM) 型 ( ϵ 2 = 0 , ≠ = {0, 1}) 方程。我们的数值模拟显示了两个和三个孤子的弹性散射，适用于各种振幅和相对速度的集合 { ϵ 1 , ϵ 2 } 的广泛值。KdV 型方程在非线性科学的几个领域中非常普遍，它们在 AdS 3 的广义相对论、玻色-爱因斯坦凝聚、超导性和孤子气体以及流体动力学中的湍流研究中找到了相关应用。分别是修正的正则化长波 (mRLW) ( ϵ 1 = ϵ 2 = 1) 和 KdV-RLW (KdV-BBM) 型 ( ϵ 2 = 0 , ≠ = {0, 1}) 方程。我们的数值模拟显示了两个和三个孤子的弹性散射，适用于各种振幅和相对速度的集合 { ϵ 1 , ϵ 2 } 的广泛值。KdV 型方程在非线性科学的几个领域中非常普遍，它们在 AdS 3 的广义相对论、玻色-爱因斯坦凝聚、超导性和孤子气体以及流体动力学中的湍流研究中找到了相关应用。