Question

The Korteweg-de Vries equation is \(u_{t}+u_{x x x}+u u_{x}=0\). Using the technique employed in computing the symmetries of the heat and wave equations, show that the infinitesimal generators of symmetries of the Korteweg-de Vries equation are $$\begin{array}{ll}\mathbf{v}_{1}=\partial_{x}, \quad \mathbf{v}_{2}=\partial_{t}, & \text { translation } \\\\\mathbf{v}_{3}=t \partial_{x}+\partial_{u}, & \text { Galilean boost } \end{array}$$ $$\mathbf{v}_{4}=x \partial_{x}+3 t \partial_{t}-2 u \partial_{u} . \quad \text { scaling }$$

Short answer

The infinitesimal generators \(\mathbf{v}_{1}=\partial_{x}\), \(\mathbf{v}_{2}=\partial_{t}\), \(\mathbf{v}_{3}=t \partial_{x}+\partial_{u}\), and \(\mathbf{v}_{4}=x \partial_{x}+3 t \partial_{t}-2 u \partial_{u}\) are valid symmetries of the Korteweg-de Vries equation.

Step-by-step solution

Write down the Korteweg-de Vries equation

The Korteweg–de Vries equation is given by \(u_{t}+u_{x x x}+u u_{x}=0\).

Write down the proposed infinitesimal generators

The proposed infinitesimal generators of symmetries are \(\mathbf{v}_{1}=\partial_{x}\), \(\mathbf{v}_{2}=\partial_{t}\), \(\mathbf{v}_{3}=t \partial_{x}+\partial_{u}\), and \(\mathbf{v}_{4}=x \partial_{x}+3 t \partial_{t}-2 u \partial_{u}\).

Verify the infinitesimal generators

We verify \(\mathbf{v}_{i}(\Phi)\) = \(L_{\mathbf{v}_{i}}(\Phi)\) = 0 \(\forall i\). Here, \(L\) denotes the Lie derivative, \(\Phi\) is the given KdV equation, and \(\mathbf{v}_{i}\) is the proposed infinitesimal generator. If all these conditions pass, then these infinitesimal generators are indeed symmetries to the Korteweg-de Vries equation.

Key concepts

Infinitesimal Generators

Infinitesimal generators are fundamental tools in understanding the symmetries of differential equations, like the Korteweg-de Vries (KdV) equation. Essentially, these generators help describe how solutions to these equations can be transformed while still remaining solutions. In simpler terms, imagine having a solution that not only works but also leads to other solutions when manipulated with these generators. For the KdV equation, the key infinitesimal generators are

• \( \mathbf{v}_{1} = \partial_{x} \) (translation in space)
• \( \mathbf{v}_{2} = \partial_{t} \) (translation in time)
• \( \mathbf{v}_{3} = t \partial_{x} + \partial_{u} \) (Galilean boost)
• \( \mathbf{v}_{4} = x \partial_{x} + 3t \partial_{t} - 2u \partial_{u} \) (scaling)
These generators allow one to explore the transformational symmetries present in the equation, providing insights into possible simplifying assumptions and invariant solutions.

Symmetries in Differential Equations

Symmetries in differential equations indicate transformations that map solutions to other solutions. Recognizing these symmetries is crucial for gaining deeper insights into the equations' nature and structure, such as with the Korteweg-de Vries equation. The KdV equation symmetries can be found using infinitesimal generators. These provide ways to change solutions systematically. For instance, translating time or space, scaling spatial features, or modifying velocity profiles. By identifying these symmetries, mathematicians can simplify equations and even discover broader classes of solutions. In turn, this understanding can lead to finding new relationships or solutions that were previously hidden.

Lie Derivatives

The Lie derivative is a comprehensive tool to analyze how a function changes along the flow generated by a vector field, like an infinitesimal generator. In the context of the KdV equation, the Lie derivative helps verify if a symmetry truly exists. When using the Lie derivative, one takes a function, like the KdV equation, and a vector field, represented by the infinitesimal generators. By computing the Lie derivative of the KdV equation with respect to each of the proposed generators, one checks if it equals zero. If the computation returns zero, this confirms that the particular transformation doesn’t change the solution set of the equation, affirming it as a symmetry. Thus, Lie derivatives are essential for proving the presence of infinitesimal symmetries in differential equations.

Partial Differential Equations

Partial Differential Equations (PDEs) are a class of equations representing functions and their partial derivatives concerning multiple variables. These equations model many natural phenomena, including the Korteweg-de Vries equation. The KdV equation specifically models waves on shallow water surfaces, making understanding its solutions and symmetries very valuable in physics and applied mathematics. Identifying symmetries through infinitesimal generators assists in finding significant solutions, which often have physical interpretations or can be used in practical applications. By breaking down PDEs into simpler symmetries and transformations, researchers can tackle complex problems more effectively, leading to a deeper understanding of the behavior of systems modeled by these equations.