Integrable partial differential equation

An integrable partial differential equation has enough exact structure, such as a Lax pair, infinitely many conserved quantities, or an inverse scattering transform, to reduce its evolution to auxiliary linear or finite-dimensional data.

The Harry Dym equation is the integrable partial differential equation $p_t=p^3{p}_{xxx}$. It admits a Lax pair based on the second-order operator $L=-{\partial }_x{p}^2{\partial }_x$.

The Boussinesq equation names a family of nonlinear dispersive wave equations. The integrable normalization used with the third-order Boussinesq Lax pair here isIts particular signs matter. Linearizing about $q=0$ and substituting $q\propto e^{\lambda t+ikx}$ givesso this normalization admits exponentially growing disturbances with arbitrarily fast growth at large wavenumber. Its formal integrable structure therefore does not establish a well-posed initial value problem in ordinary fixed-regularity function spaces. Other Boussinesq normalizations have a stabilizing fourth-derivative sign and may include a second-derivative wave term; their stability properties must be checked from their own equations.

The nonlinear term also distinguishes this equation from a collection of independent linear waves: expanding it gives $(q^2{)}_{xx}=2q_x^2+2q{q}_{xx}$. The third-order Boussinesq Lax pair supplies the specific isospectral representation for this sign convention.