▁for ▁large ▁t , ▁one ▁looks ▁for ▁asympt otic ▁solutions ▁of ▁the ▁form ▁ ▁where ▁ ▁In ser ting ▁this ▁into ▁the ▁equation , ▁one ▁finds ▁at ▁the ▁order ▁ ▁that ▁ ▁A ▁simple ▁ren aming ▁and ▁res cal ing ▁of ▁the ▁variables ▁( ass uming ▁that ▁) ▁transform s ▁this ▁into ▁the ▁Hunter – Sa xt on ▁equation . ▁ ▁General ization ▁▁ ▁The ▁analysis ▁was ▁later ▁generalized ▁by ▁Al ì ▁and ▁Hunter , ▁who ▁allowed ▁the ▁director ▁field ▁to ▁point ▁in ▁any ▁direction , ▁but ▁with ▁the ▁spatial ▁dependence ▁still ▁only ▁in ▁the ▁x ▁direction : ▁ ▁Then ▁the ▁L agr ang ian ▁is ▁ ▁The ▁corresponding ▁E uler – L ag range ▁equations ▁are ▁couple d ▁non linear ▁wave ▁equations ▁for ▁the ▁angles ▁ φ ▁and ▁ ψ , ▁with ▁ φ ▁corresponding ▁to ▁" s play ▁waves " ▁and ▁ ψ ▁to ▁" tw ist ▁waves ". ▁The ▁previous ▁Hunter – Sa xt on ▁case ▁( p ure ▁s play ▁waves ) ▁is ▁recovered ▁by ▁taking ▁ ψ ▁constant , ▁but ▁one ▁can ▁also ▁consider ▁couple d ▁s play - tw ist ▁waves ▁where ▁both ▁ φ ▁and ▁ ψ ▁vary . ▁As ym pt otic ▁exp ans ions ▁similar ▁to ▁that ▁above ▁lead ▁to ▁a ▁system ▁of ▁equations , ▁which , ▁after ▁ren aming ▁and ▁res cal ing ▁the ▁variables , ▁takes ▁the ▁form ▁ ▁where ▁u ▁is ▁related ▁to ▁ φ ▁and ▁v ▁to ▁ ψ . ▁This ▁system ▁implies ▁that ▁u ▁satisfies ▁ ▁so ▁( r