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Special exact soliton solutions for the $K(2, 2)$ equation with non-zero constant pedestal. (English) Zbl 1239.35135
Summary: Special exact solutions of the $K(2, 2)$ equation, $u_{t} + (u^{2})_{x} + (u^{2})_{xxx} = 0$, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the $K(2, 2)$ equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal $\text{lim}_{x\to \pm \infty }u = A \neq 0$, or possesses compacton solutions only when $\text{lim}_{x\to \pm \infty }u = 0$. Mathematical analysis and numerical simulations are provided for these soliton solutions of the $K(2, 2)$ equation.

35Q51Soliton-like equations
35C08Soliton solutions of PDE
Full Text: DOI
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