Effective integrable dynamics for a certain nonlinear wave equation

Abstract

We consider the following degenerate half-wave equation on the one-dimensional torus:\n¶\n<math display="block">\n<mrow>\n<mi>i</mi>\n<msub>\n<mrow>\n<mi>∂</mi>\n</mrow>\n<mrow>\n<mi>t</mi>\n</mrow>\n</msub>\n<mi>u</mi>\n<mo class="MathClass-bin">−</mo>\n<mo class="MathClass-rel">|</mo>\n<mi>D</mi>\n<mo class="MathClass-rel">|</mo>\n<mi>u</mi>\n<mo class="MathClass-rel">=</mo>\n<mo class="MathClass-rel">|</mo>\n<mi>u</mi>\n<msup>\n<mrow>\n<mo class="MathClass-rel">|</mo>\n</mrow>\n<mrow>\n<mn>2</mn>\n</mrow>\n</msup>\n<mi>u</mi>\n<mo class="MathClass-punc">,</mo>\n<mspace class="quad" width="1em"/>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mn>0</mn>\n<mo class="MathClass-punc">,</mo>\n<mo class="MathClass-bin">⋅</mo>\n<mspace class="thinspace" width="0.3em"/>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">=</mo>\n<msub>\n<mrow>\n<mi>u</mi>\n</mrow>\n<mrow>\n<mn>0</mn>\n</mrow>\n</msub>\n<mo class="MathClass-punc">.</mo>\n</mrow>\n</math>\n¶ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system—the cubic Szegő equation. As a consequence, we prove an instability result for large [math] norms of solutions of this wave equation.