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■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ♦❢ ◆❆❙ ♦❢ ❯❦r❛✐♥❡ ❚❡r❡s❤❝❤❡♥❦✐✈s❦❛ ✸✱ ❑②✐✈✱ ❯❦r❛✐♥❡

❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s✱ ♥♦r♠❛❧✐③❛t✐♦♥ ♣r♦♣❡rt② ❛♥❞ ❣r♦✉♣ ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠s ❢♦r ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

❖❧❡♥❛ ❱❛♥❡❡✈❛ ✈❛♥❡❡✈❛❅✐♠❛t❤✳❦✐❡✈✳✉❛

❳ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s ▼❡❡t✐♥❣✿ ❈♦♥❢❡r❡♥❝❡ ❛♥❞ ❙❝❤♦♦❧ ♦♥ ▼♦❞❡r♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s ❙❡♣t❡♠❜❡r ✶✸✱ ✷✵✶✾

✶

■t ✐s ✇✐❞❡❧② ❦♥♦✇♥ t❤❛t t❤❡r❡ ✐s ♥♦ ❣❡♥❡r❛❧ t❤❡♦r② ❢♦r ✐♥t❡❣r❛t✐♦♥ ♦❢ ♥♦♥❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭P❉❊s✮✳ ◆❡✈❡rt❤❡❧❡ss✱ ♠❛♥② s♣❡❝✐❛❧ ❝❛s❡s ♦❢ ❝♦♠♣❧❡t❡ ✐♥t❡❣r❛t✐♦♥ ♦r ✜♥❞✐♥❣ ♣❛rt✐❝✉❧❛r s♦❧✉t✐♦♥s ❛r❡ r❡❧❛t❡❞ t♦ ❛♣♣r♦♣r✐❛t❡ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s✳ ◆♦♥❞❡❣❡♥❡r❛t❡ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ❧❡❛✈❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥✈❛r✐❛♥t ❛♥❞ ❢♦r♠ ❛ ❝♦♥♥❡❝t❡❞ ▲✐❡ ❣r♦✉♣ ❛r❡ ❝❛❧❧❡❞ ▲✐❡ s②♠♠❡tr✐❡s ♦❢ t❤✐s ❡q✉❛t✐♦♥✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ❛r❡ ♦♥❡s ✇❤✐❝❤ ❛r❡ ♠♦st❧② ✉s❡❞✳ ❚❤✐s ♣❧❛❝❡s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❡t❤♦❞s ❛♠♦♥❣ t❤❡ ♠♦st ♣♦✇❡r❢✉❧ ❛♥❛❧②t✐❝ t♦♦❧s ❝✉rr❡♥t❧② ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ st✉❞② ♦❢ ♥♦♥❧✐♥❡❛r P❉❊s✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷

❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥♦♥tr✐✈✐❛❧ s②♠♠❡tr② ♣r♦♣❡rt✐❡s ✐s ♦♥❡ ♦❢ t❤❡ ❞✐st✐♥❝t✐✈❡ ❢❡❛t✉r❡s ✇❤✐❝❤ ❞✐✛❡r ❡q✉❛t✐♦♥s ❞❡s❝r✐❜✐♥❣ ♥❛t✉r❛❧ ♣❤❡♥♦♠❡♥❛ ❢r♦♠ ♦t❤❡r ♣♦ss✐❜❧❡ ♦♥❡s ❆❧❧ t❤❡ ❜❛s✐❝ ❡q✉❛t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✱ ✐✳❡✳ t❤❡ ❡q✉❛t✐♦♥s ♦❢ ◆❡✇t♦♥✱ ▲❛♣❧❛❝❡✱ ❞✬❆❧❡♠❜❡rt✱ ❊✉❧❡r✲▲❛❣r❛♥❣❡✱ ▲❛♠❡✱ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✱ ▼❛①✇❡❧❧✱ ❙❝❤r☎ ♦❞✐♥❣❡r ❡t❝✳✱ ❤❛✈❡ ♥♦♥tr✐✈✐❛❧ s②♠♠❡tr② ♣r♦♣❡rt✐❡s✳ ■t ♠❡❛♥s t❤❛t ♠❛♥✐❢♦❧❞s ♦❢ t❤❡✐r s♦❧✉t✐♦♥s ❛r❡ ✐♥✈❛r✐❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ♠✉❧t✐✲♣❛r❛♠❡t❡r ❣r♦✉♣ ♦❢ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥s ✭▲✐❡ ❣r♦✉♣ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s✮ ✇✐t❤ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✸

❚❤❡ r❡q✉✐r❡♠❡♥t ♦❢ ✐♥✈❛r✐❛♥❝❡ ♦❢ ❛♥ ❡q✉❛t✐♦♥ ✉♥❞❡r ❛ ❣r♦✉♣ ❡♥❛❜❧❡s ✉s ✐♥ s♦♠❡ ❝❛s❡s t♦ s❡❧❡❝t t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ❛ ✇✐❞❡ s❡t ♦❢ ♦t❤❡r ❛❞♠✐ss✐❜❧❡ ♦♥❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡r❡ ✐s t❤❡ ♦♥❧② ♦♥❡ s②st❡♠ ♦❢ P♦✐♥❝❛r✁ ❡✲✐♥✈❛r✐❛♥t ✜rst✲♦r❞❡r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ❢♦r t✇♦ r❡❛❧ ✈❡❝t♦rs E ❛♥❞ H✱ ❛♥❞ t❤✐s ✐s t❤❡ s②st❡♠ ♦❢ ▼❛①✇❡❧❧ ❡q✉❛t✐♦♥s✳

• r♦✉♣ ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❝♦♥s✐sts ✐♥ t❤❡

s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ♥♦♥✲❡q✉✐✈❛❧❡♥t ❝❛s❡s ♦❢ s✉❝❤ ❡q✉❛t✐♦♥s ✇❤✐❝❤ ♣♦ss❡ss t❤❡ ❡①t❡♥s✐♦♥s ♦❢ ▲✐❡ s②♠♠❡tr✐❡s✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✹

❚❤❡ s②st❡♠❛t✐❝ st✉❞② ♦❢ tr❛♥s❢♦r♠❛t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝❧❛ss❡s ♦❢ ♥♦♥❧✐♥❡❛r P❉❊s ✇❛s ✐♥✐t✐❛t❡❞ ✐♥ ✶✾✾✶ ❜② ❑✐♥❣st♦♥ ❛♥❞ ❙♦♣❤♦❝❧❡♦✉s✳ ❚❤❡s❡ ❛✉t❤♦rs ❧❛t❡r ♥❛♠❡❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s r❡❧❛t❡❞ t✇♦ ♣❛rt✐❝✉❧❛r ❡q✉❛t✐♦♥s ✐♥ ❛ ❝❧❛ss ♦❢ P❉❊s ❢♦r♠✲♣r❡s❡r✈✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❜❡❝❛✉s❡ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s ♣r❡s❡r✈❡ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐♥ ❛ ❝❧❛ss ❛♥❞ ❝❤❛♥❣❡ ♦♥❧② ✐ts ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✳ ❖♥❧② ❛ ②❡❛r ❧❛t❡r ✐♥ ✶✾✾✷ ●❛③❡❛✉ ❛♥❞ ❲✐♥t❡r♥✐t③ st❛rt❡❞ t♦ ✐♥✈❡st✐❣❛t❡ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ ❝❧❛ss❡s ♦❢ P❉❊s ❝❛❧❧✐♥❣ t❤❡♠ ❛❧❧♦✇❡❞ tr❛♥s❢♦r♠❛t✐♦♥s✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✺

❘✐❣♦r♦✉s ❞❡✜♥✐t✐♦♥s ❛♥❞ ❞❡✈❡❧♦♣❡❞ t❤❡♦r② ♦♥ t❤❡ s✉❜❥❡❝t ✇❛s ♣r♦♣♦s❡❞ ❧❛t❡r ❜② P♦♣♦✈②❝❤✳ ❆s ❢♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♥♦t✐♦♥ ♦❢ ❢♦r♠✲♣r❡s❡r✈✐♥❣ ✭❛❧❧♦✇❡❞✮ tr❛♥s❢♦r♠❛t✐♦♥s ❤❡ s✉❣❣❡st❡❞ t❤❡ t❡r♠ ❛❞♠✐ss✐❜❧❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ■♥ ❜r✐❡❢✱ ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ tr✐♣❧❡ ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ✜①❡❞ ❡q✉❛t✐♦♥s ❢r♦♠ ❛ ❝❧❛ss ❛♥❞ ❛ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ❧✐♥❦s t❤❡s❡ ❡q✉❛t✐♦♥s✳ ❚❤❡ s❡t ♦❢ ❛❞♠✐ss✐❜❧❡ tr❛♥s❢♦r♠❛t✐♦♥s ❝♦♥s✐❞❡r❡❞ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ♦♣❡r❛t✐♦♥ ♦❢ ❝♦♠♣♦s✐t✐♦♥ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✻

❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣

❚❤❡r❡ ❡①✐st s❡✈❡r❛❧ ❦✐♥❞s ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣s ❞❡♣❡♥❞✐♥❣ ♦♥ r❡str✐❝t✐♦♥s t❤❛t ❛r❡ ✐♠♣♦s❡❞ ♦♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤❡ ✉s✉❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ♦❢ t❤❡ ❝❧❛ss L|S ❝♦♥s✐sts ♦❢ t❤❡ ♥♦♥❞❡❣❡♥❡r❛t❡ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥s ✐♥ t❤❡ s♣❛❝❡ ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✱ ✇❤✐❝❤ ♣r❡s❡r✈❡ t❤❡ ✇❤♦❧❡ ❝❧❛ss L|S ❛♥❞ ❛r❡ ♣r♦❥❡❝t❛❜❧❡ ♦♥ t❤❡ ✈❛r✐❛❜❧❡ s♣❛❝❡✱ ✐✳❡✳✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✼

• ❡♥❡r❛❧✐③❡❞ ❡①t❡♥❞❡❞ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣

❘❡str✐❝t✐♦♥s ♦♥ tr❛♥s❢♦r♠❛t✐♦♥s ❝❛♥ ❜❡ ✇❡❛❦❡♥❡❞ ✐♥ t✇♦ ❞✐r❡❝t✐♦♥s✳ ❲❡ ❛❞♠✐t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s t✱ x ❛♥❞ u ❝❛♥ ❞❡♣❡♥❞ ♦♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts ✭t❤❡ ♣r❡✜① ✏❣❡♥❡r❛❧✐③❡❞✑ ❬▼❡❧❡s❤❦♦✱ ✶✾✾✹❪✮✳ ❚❤❡ ❡①♣❧✐❝✐t ❢♦r♠ ♦❢ t❤❡ ♥❡✇ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts (˜ f, ˜ g, ˜ h, ˜ n, ˜ m) ✐s ❞❡t❡r♠✐♥❡❞ ✈✐❛ (t, x, u, f, g, h, n, m) ✐♥ s♦♠❡ ♥♦♥✲✜①❡❞ ✭♣♦ss✐❜❧②✱ ♥♦♥❧♦❝❛❧✮ ✇❛② ✭t❤❡ ♣r❡✜① ✏❡①t❡♥❞❡❞✑ ❬■✈❛♥♦✈❛✫P♦♣♦✈②❝❤✫❙♦♣❤♦❝❧❡♦✉s✱ ✷✵✵✺❪✮✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✽

◆♦r♠❛❧✐③❛t✐♦♥ ♣r♦♣❡rt②

❚❤❡ ❝❧❛ss L|S ✐s ❝❛❧❧❡❞ ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ ✭r❡s♣✳ ❣❡♥❡r❛❧✐③❡❞✱ r❡s♣✳ ❡①t❡♥❞❡❞✱ r❡s♣✳ ❣❡♥❡r❛❧✐③❡❞ ❡①t❡♥❞❡❞✮ s❡♥s❡ ✐❢ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞ ♦❢ t❤✐s ❝❧❛ss ✐s ✐♥❞✉❝❡❞ ❜② tr❛♥s❢♦r♠❛t✐♦♥s ❢r♦♠ ✐ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t②♣❡✳ ❈❧❛ss❡s ✇❤✐❝❤ ❛r❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ❢♦r ✐♥✈❡st✐❣❛t✐♦♥ ❛r❡ t❤♦s❡ ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ❈❧❛ss ♥♦r♠❛❧✐③❡❞ ✐♥ ❛♥② s❡♥s❡ ✐s ❛❧✇❛②s ❜❡tt❡r t❤❛♥ ♦♥❡ t❤❛t ✐s ♥♦t ♥♦r♠❛❧✐③❡❞✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✾

❈♦♥s✐❞❡r t❤❡ ❣❡♥❡r❛❧ ❝❧❛ss ♦❢ s❡❝♦♥❞✲♦r❞❡r ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✱ ut = H(t, x, u, ux, uxx), Huxx = 0. ❆♥② ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ T r❡❧❛t✐♥❣ t✇♦ ✜①❡❞ ❡q✉❛t✐♦♥s ut = H ❛♥❞ ˜ u˜

t = ˜

H ❢r♦♠ t❤✐s ❝❧❛ss ❤❛s t❤❡ ❢♦r♠ ˜ t = T(t)✱ ˜ x = X(t, x, u)✱ ˜ u = U(t, x, u) ✇✐t❤ Tt(XxUu − XuUx) = 0✳ ˜ u˜

t = DtUDxX − DxUDtX

TtDxX , ˜ u˜

x = DxU

DxX , ˜ u˜

x˜ x =

1 DxX Dx DxU DxX

• ,

✇❤❡r❡ Dt = ∂t + ut∂u + utt∂ut + utx∂ux + . . . ❛♥❞ Dx = ∂x + ux∂u + utx∂ut + uxx∂ux + . . .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✵

Pr♦♣♦s✐t✐♦♥ ✶✳

❚❤❡ ❝❧❛ss ut = H(t, x, u, ux, uxx), Huxx = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ✐s ❢♦r♠❡❞ ❜② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x, u), ˜ u = U(t, x, u), Tt(XxUu − XuUx) = 0, ˜ H = XxUu − XuUx TtDxX H + UtDxX − XtDxU TtDxX . ❚❤❡ s✉❜❝❧❛ss ♦❢ t❤❡ ❛❜♦✈❡ ❝❧❛ss s✐♥❣❧❡❞ ♦✉t ❜② t❤❡ ❝♦♥❞✐t✐♦♥ Huxxuxx = 0 ❤❛s t❤❡ s❛♠❡ ❡q✉✐✈❛❧❡♥❝❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ❢♦r ✈❛r✐❛❜❧❡s✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✶

Pr♦♣♦s✐t✐♦♥ ✷✳

❚❤❡ ❝❧❛ss ♦❢ q✉❛s✐❧✐♥❡❛r s❡❝♦♥❞✲♦r❞❡r ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✱ ut = G(t, x, u, ux)uxx + F(t, x, u, ux), G = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ✐s ❢♦r♠❡❞ ❜② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x, u), ˜ u = U(t, x, u), Tt(XxUu − XuUx) = 0, ˜ G = (DxX)2 Tt G, ˜ F = XxUu − XuUx TtDxX F + UtDxX − XtDxU TtDxX + (Xxx + 2Xxuux + Xuuu2

x)DxU − (Uxx + 2Uxuux + Uuuu2 x)DxX

TtDxX G.

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✷

Pr♦♣♦s✐t✐♦♥ ✸✳

❚❤❡ ❝❧❛ss ut = G(t, x, u)uxx + F(t, x, u, ux), G = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♠♣r✐s❡s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U(t, x, u), ˜ G = X 2

x

Tt G, ˜ F = Uu Tt F + UtXx − XtDxU TtXx + XxxDxU − (Uxx + 2Uxuux + Uuuu2

x)Xx

TtXx G, ✇❤❡r❡ TtXxUu = 0✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✸

Pr♦♣♦s✐t✐♦♥ ✹✳

❚❤❡ ❝❧❛ss ut = G(t, x, u)uxx + n

k=0 F k(t, x, u)uk x , n ≥ 2,

G = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♥s✐sts ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U(t, x, u), ˜ G = X 2

x

Tt G ❛♥❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ❢♦r t❤❡ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts F k✱ k = 0, . . . , n, ❛r❡ ❢♦✉♥❞ ❛s s♦❧✉t✐♦♥s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ s②st❡♠ r❡s✉❧t✐♥❣ ❢r♦♠ t❤❡ s♣❧✐tt✐♥❣ ✇✳r✳t✳ ❞✐✛❡r❡♥t ♣♦✇❡rs ♦❢ ux ♦❢ t❤❡ ❡q✉❛t✐♦♥

n

• k=0

˜ F k Uu Xx ux + Ux Xx k = 1 TtXx

• XxUu

n

• k=0

F kuk

x + UtXx − XtDxU +

• XxxDxU − (Uxx + 2Uxuux + Uuuu2

x)Xx

• G
• .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✹

Pr♦♣♦s✐t✐♦♥ ✺✳

❚❤❡ ❝❧❛ss ut = G(t, x, u)uxx + F 2(t, x, u)u2

x + F 1(t, x, u)ux + F 0(t, x, u),

G = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♥s✐sts ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U(t, x, u), ˜ G = X 2

x

Tt G, ˜ F 2 = X 2

x

TtU2

u

• UuF 2 − UuuG
• , ˜

F 1 = 1 TtUu

• 2XxUx

Uu (UuuG − UuF 2) + XxUuF 1 − XtUu + (XxxUu − 2UxuXx)G

• ,

˜ F 0 = 1 Tt U2

x

Uu F 2 − UxF 1 + UuF 0 + Ut +

• 2Ux

Uu Uxu − Uxx − U2

x

U2

u

Uuu

• G
• ,

✇❤❡r❡ TtXxUu = 0.

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✺

Pr♦♣♦s✐t✐♦♥ ✻✳

❚❤❡ ❝❧❛ss ut = G(t, x, u)uxx + F 1(t, x, u)ux + F 0(t, x, u), G = 0, ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♠♣r✐s❡s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U1(t, x)u + U0(t, x), TtXxU1 = 0, ˜ G = X 2

x

Tt G, ˜ F 1 = 1 TtU1

• XxU1F 1 − XtU1 + (XxxU1 − 2U1

x Xx)G

• ,

˜ F 0 = 1 Tt

• U1F 0 − (U1

x u + U0 x )F 1 + U1 t u + U0 t +

+

• 2U1

x

U1 (U1

x u + U0 x ) − U1 xxu − U0 xx

• G
• .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✻

❈♦♥s✐❞❡r ♦♥❡ ♠♦r❡ s✉❜❝❧❛ss ♦❢ t❤❡ ❝❧❛ss ut = G(t, x, u)uxx + F 2(t, x, u)u2

x + F 1(t, x, u)ux + F 0(t, x, u),

✭✶✮ ❢♦r ✇❤✐❝❤ t❤❡ ❝♦♥❞✐t✐♦♥ Uuu = 0 ❤♦❧❞s ❢♦r ❛❞♠✐ss✐❜❧❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤✐s ✐s t❤❡ s✉❜❝❧❛ss s✐♥❣❧❡❞ ♦✉t ❜② t❤❡ ❝♦♥❞✐t✐♦♥ F 2 = Gu✱ ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u), G = 0. ❚❤✐s ❝❧❛ss ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ut = Guxx + Guu2

x + (Gx + K)ux + P✱ ✇❤❡r❡ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥

❛r❜✐tr❛r② ❡❧❡♠❡♥ts ♦❢ t❤❡ ❧❛tt❡r ❝❧❛ss ❛♥❞ ❝❧❛ss ✭✶✮ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛s F 2 = Gu✱ F 1 = Gx + K✱ F 0 = P✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✼

Pr♦♣♦s✐t✐♦♥ ✼✳

❚❤❡ ❝❧❛ss ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u) ✐s ♥♦r♠❛❧✐③❡❞ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ■ts ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ✐s ❢♦r♠❡❞ ❜② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U1(t, x)u + U0(t, x), TtXxU1 = 0, ˜ G = X 2

x

Tt G, ˜ K = Xx Tt

• K −

Xxx Xx + 2U1

x

U1

• G − 2(U1

x u + U0 x )Gu

U1 − Xt Xx

• ,

˜ P = 1 Tt

• U1P + (U1

x u + U0 x )2

U1 Gu − (U1

x u + U0 x )(Gx + K) + U1 t u + U0 t +

• 2U1

x

U1 (U1

x u + U0 x ) − U1 xxu − U0 xx

• G
• .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✽

❚❤❡ ❝❧❛ss ut = (G(t, x, u)ux)x + P(t, x, u), G = 0, ✭✷✮ ✐s ♥♦t ♥♦r♠❛❧✐③❡❞ ❛♥②♠♦r❡✳ ■t✬s ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ✐s ❝♦♠♣r✐s❡❞ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = δ1x + δ2, ˜ u = U1(t)u + U0(t), TtU1δ1 = 0, ˜ G = δ2

1

Tt G, ˜ P = 1 Tt

• U1P + U1

t u + U0 t

• .

■❢ G ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ (au + b)Gu + c G + d = 0, t❤❡♥ ❝❧❛ss ✭✷✮ ✐s ♥♦r♠❛❧✐③❡❞✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✶✾

❲❡ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ ❝❧❛ss S(t, x)ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u), SG = 0. ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❝❧❛ss❡s ♦❢ ✈❛r✐❛❜❧❡✲❝♦❡✣❝✐❡♥t ❞✐✛✉s✐♦♥✕r❡❛❝t✐♦♥ ❡q✉❛t✐♦♥s f(x)ut = (g(x)A(u)ux)x + h(x)B(u) ❛♥❞ ❞✐✛✉s✐♦♥✕❝♦♥✈❡❝t✐♦♥ ❡q✉❛t✐♦♥s f(x)ut = (g(x)A(u)ux)x + h(x)B(u)ux ❛r❡ s✉❜❝❧❛ss❡s ♦❢ t❤✐s ❝❧❛ss✳ ❚❤❡ ❝♦❡✣❝✐❡♥t S(t, x) ❝❛♥ ❜❡ ❣❛✉❣❡❞ t♦ ♦♥❡ ❜② t❤❡ ❢❛♠✐❧② ♦❢ ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ ˜ t = t, ˜ x = x

x0

S(t, y) ❞y, ˜ u = u. ◆❡✈❡rt❤❡❧❡ss✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤✐s ❝❧❛ss s❡♣❛r❛t❡❧② s✐♥❝❡ ✐ts tr❛♥s❢♦r♠❛t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ❜❡❝♦♠❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✵

Pr♦♣♦s✐t✐♦♥ ✽✳ ❆♥② ♣♦✐♥t tr❛♥s❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❡q✉❛t✐♦♥s ❢r♦♠ t❤❡ ❝❧❛ss S(t, x)ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u) ❤❛s t❤❡ ❢♦r♠ ˜ t = T(t), ˜ x = X(t, x), ˜ u = U1(t, x)u + U0(t, x), TtXxU1 = 0. ❚❤❡♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts ❛r❡ r❡❧❛t❡❞ ✈✐❛ t❤❡ ❢♦r♠✉❧❛s ˜ G ˜ S = X 2

x

Tt

G S ,

˜ K + ˜ G˜

x

˜ S = Xx TtS

• K + Gx +

Xxx Xx − 2U1

x

U1

• G − 2(U1

x u + U0 x )Gu

U1 − Xt Xx S

• ,

˜ P ˜ S = 1 TtS

• U1P + (U1

x u + U0 x )2

U1 Gu − (U1

x u + U0 x )(K + Gx) + (U1 t u + U0 t )S +

• 2U1

x

U1 (U1

x u + U0 x ) − U1 xxu − U0 xx

• G
• .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✶

■t ✐s ♦❜✈✐♦✉s t❤❛t tr❛♥s❢♦r♠❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝❧❛ss S(t, x)ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u), SG = 0. ❜❡❝♦♠❡ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤♦s❡ ♦❢ ✐ts s✉❜❝❧❛ss ✇✐t❤ S = 1✳ ❚r❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ♦♥❧② ❢♦r ❢r❛❝t✐♦♥s ♦❢ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✳ ■t ✐s ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤✐s ❝❧❛ss ❛❞♠✐ts ♣❡❝✉❧✐❛r ❣❛✉❣❡ ❡q✉✐✈❛❧❡♥❝❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭❛♥ ❡q✉✐✈❛❧❡♥❝❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r ✇❤✐❝❤ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❞♦ ♥♦t tr❛♥s❢♦r♠ ❜✉t ♦♥❧② ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✮✳ ❚❤✐s ✐s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ˜ S = Z(t, x, S), ˜ G = G S Z, ˜ K = K S Z − G Z S

• x

, ˜ P = P S Z, ✇❤❡r❡ Z ✐s ❛♥ ❛r❜✐tr❛r② s♠♦♦t❤ ❢✉♥❝t✐♦♥ ♦❢ ✐ts ✈❛r✐❛❜❧❡s ✇✐t❤ ZS = 0✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✷

Pr♦♣♦s✐t✐♦♥ ✾✳ ❚❤❡ ❝❧❛ss S(t, x)ut = (G(t, x, u)ux)x + K(t, x, u)ux + P(t, x, u) ✐s ♥♦r♠❛❧✐③❡❞✳ ■t✬s ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♠♣r✐s❡s t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X(t, x), ˜ u = U1(t, x)u + U0(t, x), TtXxU1 = 0, ˜ S = Z(t, x, S), ˜ G = X 2

x

Tt G S Z, ˜ K = XxZ TtS

• K −

Xxx Xx + 2U1

x

U1

• G − 2(U1

x u + U0 x )Gu

U1 − Xt Xx S

• − Xx

Tt G Z S

• x

, ˜ P = Z TtS

• U1P + (U1

x u + U0 x )2

U1 Gu − (U1

x u + U0 x )(K + Gx) + (U1 t u + U0 t )S +

• 2U1

x

U1 (U1

x u + U0 x ) − U1 xxu − U0 xx

• G
• .

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✸

❱❛♥❡❡✈❛ ❖✳❖✳ ❡t ❛❧✳ ✭✷✵✶✹✮ P❤②s✐❝❛ ❙❝r✐♣t❛✱ ✽✾ ✭✸✮✱ ✵✸✽✵✵✸✳

❚❤❡ ❝❧❛ss ut = F(t)un + G(t, x, u0, u1, . . . , un−1), F = 0, Guiun−1 = 0, ✇❤❡r❡ i = 1, . . . , n − 1, n 2✱ ✐s ♥♦r♠❛❧✐③❡❞✳ ■ts ✉s✉❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ ❝♦♥s✐sts ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = X 1(t)x +X 0(t), ˜ u = U1(t, x)u+U0(t, x), ˜ F = (X 1)n Tt F, ˜ G = U1 Tt G − n−1

• k=0
• n

k

• U1

n−kuk + U0 n

• F

Tt + U1

t

Tt u+ +U0

t

Tt − X 1

t x + X 0 t

TtX 1 (U1ux + U1

x u + U0 x ),

✇❤❡r❡ TtX 1U1 = 0✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✹

• ❡♥❡r❛❧✐③❡❞ ❑❛✇❛❤❛r❛ ❡q✉❛t✐♦♥s

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❖✳ ▼❛❣❞❛ ❛♥❞ ❆✳ ❩❤❛❧✐❥

❲❡ st✉❞② ❣❡♥❡r❛❧✐③❡❞ ❑❛✇❛❤❛r❛ ❡q✉❛t✐♦♥s ut + α(t)f(u)ux + β(t)uxxx + σ(t)uxxxxx = 0, ✭✸✮ ❢r♦♠ t❤❡ ▲✐❡ s②♠♠❡tr② ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❍❡r❡ f✱ α✱ β ❛♥❞ σ ❛r❡ s♠♦♦t❤ ♥♦♥✈❛♥✐s❤✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ t❤❡✐r ✈❛r✐❛❜❧❡s✳ ❚❤❡ ❝❧❛ss ✭✸✮ ✐s ♥♦t ♥♦r♠❛❧✐③❡❞ ❜✉t ✐t ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ t✇♦ ♥♦r♠❛❧✐③❡❞ s✉❜❝❧❛ss❡s ✇❤✐❝❤ ❛r❡ s✐♥❣❧❡❞ ♦✉t ❜② t❤❡ ❝♦♥❞✐t✐♦♥s✱ fuu = 0 ❛♥❞ fuu = 0, r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❝❛s❡ f(u) = un, n = 0 ✐s ✐♥✈❡st✐❣❛t❡❞ ❡①❤❛✉st✐✈❡❧② ✐♥ ❬❖✳ ❑✉r✐❦s❤❛✱ ❙✳ P♦✞ st❛✱ ❖✳ ❱❛♥❡❡✈❛✱ ❏✳ P❤②s✳ ❆✿ ▼❛t❤✳ ❚❤❡♦r✳ ✹✼ ✭✷✵✶✹✮ ✵✹✺✷✵✶❪✳ ❖t❤❡r r❡❧❛t❡❞ ✇♦r❦s✿ ❬▼✳▲✳ ●❛♥❞❛r✐❛s✱ ▼✳ ❘♦s❛✱ ❊✳ ❘❡❝✐♦✱ ❙✳❈✳ ❆♥❝♦✱ ❆■P ❈♦♥❢❡r❡♥❝❡ Pr♦❝❡❡❞✐♥❣s ✶✽✸✻ ✭✷✵✶✼✮✱ ✵✷✵✵✼✷❪✳ ❬❏✳ ❱❛✞ s✐✞ ❝❡❦✱ ❛r❳✐✈✿✶✽✶✵✳✵✷✽✻✸❪✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✺

• ❡♥❡r❛❧✐③❡❞ ❑❛✇❛❤❛r❛ ❡q✉❛t✐♦♥s

❚❤❡♦r❡♠ ✶✳

❚❤❡ ✉s✉❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣ G∼ ♦❢ ❝❧❛ss ut + α(t)f(u)ux + β(t)uxxx + σ(t)uxxxxx = 0 ❝♦♥s✐sts ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ˜ t = T(t), ˜ x = δ1x + δ2

• a(t)❞t + δ3,

˜ u = δ4u + δ5, ˜ f = δ0

• f + δ2

δ1

• ,

˜ α(˜ t) = δ1 δ0Tt α(t), ˜ β(˜ t) = δ3

1

Tt β(t), ˜ σ(˜ t) = δ5

1

Tt σ(t), ✇❤❡r❡ δj, j = 0, 1, 2, 3, 4, 5 ❛r❡ ❛r❜✐tr❛r② ❝♦♥st❛♥ts ✇✐t❤ δ0δ1δ3 = 0✱ T ✐s ❛♥ ❛r❜✐tr❛r② s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✇✐t❤ Tt = 0.

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✻

❚❛❜❧❡ ✶✳ ❚❤❡ ❣r♦✉♣ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ut + f(u)ux + β(t)uxxx + σ(t)uxxxxx = 0✱ βσ = 0✱ f = un✳ f(u) β(t) σ(t) ❇❛s✐s ♦❢ Amax ✵ ∀ ∀ ∀ ∂x ✶ ∀ λt2 δt4 ∂x, t∂t + x∂x ✷ ∀ λ δ ∂x, ∂t ✸ eu λtρ δt

5ρ+2 3

∂x, 3t∂t + (ρ + 1)x∂x + (ρ − 2)∂u ✹ eu λet δe

5 3 t

∂x, 3∂t + x∂x + ∂u ✺ ln u ∀ ∀ ∂x, t∂x + u∂u ✻ ln u λ δ ∂x, ∂t, t∂x + u∂u

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✼

ut + unux + β(t)uxxx + σ(t)uxxxxx = 0

β(t) σ(t) ❇❛s✐s ♦❢ Amax n = 1 ∀ ∀ ∂x λtρ δt

5ρ+2 3

∂x, 3nt∂t + (ρ + 1)nx∂x + (ρ − 2)u∂u λet δe

5 3 t

∂x, 3n∂t + nx∂x + u∂u λ δ ∂x, ∂t n = 1 ∀ ∀ ∂x, t∂x + ∂u λtρ δt

5ρ+2 3

∂x, t∂x + ∂u, 3t∂t + (ρ + 1)x∂x + (ρ − 2)u∂u λet δe

5 3 t

∂x, t∂x + ∂u, 3∂t + x∂x + u∂u λ δ ∂x, t∂x + ∂u, ∂t λ(t2 + 1)

1 2 e3ν arctan t

δ(t2 + 1)

3 2 e5ν arctan t

∂x, t∂x + ∂u, (t2 + 1)∂t+ (t + ν)x∂x + ((ν − t)u + x)∂u

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✽

❈♦♥❝❧✉s✐♦♥s

◮ ❖♥❝❡ ✐t ✐s ♣r♦✈❡❞ t❤❛t s♦♠❡ ❝❧❛ss ✐s ♥♦r♠❛❧✐③❡❞ t❤❡ ✜♥❞✐♥❣ ♦❢ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞ ❢♦r ✐ts s✉❜❝❧❛ss❡s ❜❡❝♦♠❡s ❡ss❡♥t✐❛❧❧② s✐♠♣❧❡r✳ ◮ ❚❤❡ st✉❞② ♦❢ tr❛♥s❢♦r♠❛t✐♦♥❛❧ ❛♥❞ ♥♦r♠❛❧✐③❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ❝❧❛ss❡s ♦❢ P❉❊s ❝❛♥ s✐♠♣❧✐❢② ❛ ❧♦t t❤❡ ❢✉rt❤❡r st✉❞② ♦❢ t❤❡✐r s②♠♠❡tr② ♣r♦♣❡rt✐❡s✳ ❋♦r ❝❧❛ss❡s t❤❛t ❛r❡ ♥♦t ♥♦r♠❛❧✐③❡❞ t❤❡ ♠❡t❤♦❞ ♦❢ ♣❛rt✐t✐♦♥ ♦❢ ❛ ❝❧❛ss ✐♥t♦ ♥♦r♠❛❧✐③❡❞ s✉❜❝❧❛ss❡s ✇♦r❦s ✈❡r② ❣♦♦❞✳ ◮ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ♠❛♥② ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ ❝❧❛ss❡s ♦❢ ❉❊s✿ ✜♥❞✐♥❣ ❡①❛❝t s♦❧✉t✐♦♥s ❛♥❞ ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s✱ st✉❞② ♦❢ t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❛♥❞ ♠♦r❡✳

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s

✷✾

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❱❛♥❡❡✈❛ ❖✳ ⑤ ❊q✉✐✈❛❧❡♥❝❡ ❣r♦✉♣♦✐❞s ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s