[Paper Review] The group classification of one class of nonlinear wave equations

V. Lahno, Olena Magda|ArXiv.org|Oct 23, 2003

Nonlinear Waves and Solitons10 references3 citations

TL;DR

This paper presents a complete group classification of a class of nonlinear hyperbolic wave equations in (1+1) dimensions, specifically $ u_{tx} = g(t,x)u_x + f(t,x,u) $ and $ u_{tx} = f(t,x,u) $, using symmetry analysis and equivalence group methods. The key contribution is the full classification of all possible Lie symmetry algebras—up to dimension three—along with the corresponding invariant forms of $ f $ and $ g $, providing a complete list of inequivalent equations admitting nontrivial symmetries.

ABSTRACT

The problem of group classification of one class of quasilinear equations of hyperbolic type with two independent variables has been solved completely.

Motivation & Objective

• To solve the long-standing problem of group classification for a specific class of quasilinear hyperbolic wave equations with two independent variables.
• To determine all possible Lie symmetry algebras that can be realized by the equations $ u_{tx} = g(t,x)u_x + f(t,x,u) $ and $ u_{tx} = f(t,x,u) $, under the condition $ f_{uu} \neq 0 $.
• To classify all inequivalent equations admitting nontrivial point symmetries by constructing canonical forms of symmetry operators using the equivalence group.
• To provide a complete list of all possible realizations of one- and two-dimensional Lie algebras as symmetry algebras of the equations, including solvable and semisimple cases.
• To establish the maximal possible symmetry algebras for the class, showing that only algebras of dimension ≤3 are possible, and to derive explicit functional forms of $ f $ and $ g $ for each case.

Proposed method

• Employing the Lie symmetry method to derive the determining system for point symmetry generators of the form $ Q = \tau(t,x,u)\partial_t + \xi(t,x,u)\partial_x + \eta(t,x,u)\partial_u $.
• Using the equivalence group of transformations $ \bar{t} = T(t), \bar{x} = X(x), \bar{u} = U(t,x,u) $ to reduce the general symmetry operator to canonical forms, simplifying the classification.
• Applying the method of canonical forms of linear partial differential operators (LPDOs) under the equivalence group to classify symmetry algebras systematically.
• Analyzing the structure of the symmetry algebra by assuming possible Lie algebra types (e.g., $ A_{2.1}, A_{2.2}, sl(2,\mathbb{R}) $) and solving the resulting system of PDEs for $ f $ and $ g $.
• Using the condition $ f_{uu} \neq 0 $ to ensure nonlinearity and avoid degeneracy, and focusing on local solutions with smooth functions.
• Deriving explicit functional forms of $ f $ and $ g $ for each symmetry algebra realization, including self-similar and exponential forms, via substitution and change of variables.

Experimental results

Research questions

• RQ1 What are all the possible Lie symmetry algebras that can be realized by the class of nonlinear hyperbolic wave equations $ u_{tx} = g(t,x)u_x + f(t,x,u) $ with $ f_{uu} \neq 0 $?
• RQ2 How can the equivalence group of transformations be used to classify symmetry operators into canonical forms and simplify the group classification problem?
• RQ3 What are the explicit functional forms of $ f $ and $ g $ corresponding to each inequivalent realization of a given Lie algebra as a symmetry algebra?
• RQ4 Which symmetry algebras (e.g., abelian, solvable, semisimple) can occur, and what are the maximal dimensions of such algebras for this class of equations?
• RQ5 How do the invariant forms of $ f $ and $ g $ depend on the structure of the symmetry algebra, particularly for $ A_{2.2} $, $ sl(2,\mathbb{R}) $, and $ A_{3.6} $?

Key findings

• The symmetry algebra of equation (1.1) is generated by operators of the form $ Q = \tau(t)\partial_t + \xi(x)\partial_x + (h(t)u + r(t,x))\partial_u $, with $ \tau, \xi, h, r $ satisfying a system of PDEs involving $ f $ and $ g $.
• For equation (1.1), there are three inequivalent realizations of one-dimensional Lie algebras as symmetry algebras, corresponding to different functional forms of $ f $ and $ g $.
• There are three inequivalent realizations of the two-dimensional solvable algebra $ A_{2.2} $, each corresponding to distinct functional forms: $ g = [mt + (k-m)x]t^{-1}(t-x)^{-1} $, $ g = t^{-2}(kx + mt) $, and $ g = (tx)^{-1}(mx - t) $, with $ f $ expressed in terms of self-similar variables.
• For equation (1.2), the complete classification yields nine inequivalent cases, including two with $ A_{2.1} $, two with $ A_{2.2} $, one with $ sl(2,\mathbb{R}) $, and others with $ A_{2.2} \oplus A_1 $ and $ A_{2.2} \oplus A_{2.2} $, each with specific $ f $-forms such as $ f = e^t \tilde{f}(\omega) $ or $ f = (t-x)^{-3} \tilde{f}(\omega) $.
• The maximal symmetry algebra for the class is $ A_{3.6} $, realized when $ f = \tilde{f}(u) $, with symmetry generators $ \partial_t, \partial_x, -t\partial_t - x\partial_x $, and $ f_{uu} \neq 0 $, indicating a high degree of invariance.
• The paper establishes that all possible symmetry algebras are of dimension at most three, and no higher-dimensional algebras exist for this class of equations.