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Similarity solutions of the generalized Korteweg-de Vries equation. (English) Zbl 0939.35164

Numerical simulations and group invariance considerations suggest the existence of similarity solutions of the form \[ U(x,t)=\frac{1}{(t_*-t)^{\frac{2}{3}p}}\psi \Biggl(\frac{x^*-x-c(t_*-t)^{\frac{1}{3}}} {(t_*-t)^{\frac{1}{3}}}\Biggr) \] of the generalized Korteweg-de Vries equation \(u_t+u^pu_x+u_{xxx}=0\) that recently has arisen as a model for waves in a variety of nonlinear dispersive media, respectively in modelling waves in a crystalline lattice for \(p=2\).
If \(U(x,t)=\frac{1}{(t_*-t)^{\frac{2}{3}p}}\Psi(\frac{x^*-x-c(t_*-t)^{\frac{1}{3}}} {(t_*-t)^{\frac{1}{3}}})+\) bounded term , with \(x_*,t_*\) and \(c\) real parameters, ignoring the bounded remainder, \(\Psi\) would have to satisfy the ordinary differential equation \(\Psi'''+\Psi^p \Psi'-\frac{2}{3p}\Psi -\) \(-\frac{1}{3}(z+c)\Psi'=0\) where \(z=\frac{x_*-x}{(t_*-t)^{\frac{1}{3}}}-c\) connotes the similarity variable.
If \(s=z+c\) and \( \varphi(s)=\Psi(s-c)\), then \(\varphi\) would satisfy a similar equation with \(s\) as independent variable.
The main result is given by the following theorem.
There exists an infinite family of non-trivial, \(C^\infty\) solutions \(\varphi\) with \(p=4\), defined on the entire real axis and having the following properties:
(i) \(\varphi(s)>0\) for \(s>0\) and \(\varphi(s)=cs^{-1/2} \exp(-2s^{3/2}/3\sqrt{3})(1-\frac{2}{3\sqrt{3}s^{3/2}}+o(s^{-3/2}))\) as \(s \rightarrow \infty\), where \(c\) is a positive constant, and
(ii) there exist real constants \(a\) and \(b\), not both zero, such that \[ \varphi(s) = (-s)^{-1/2}[a \cos((-s)^{3/2}/3\sqrt{3})+b \sin((-s)^{3/2}/3\sqrt{3})+O(-s)^{-3/2}]^2 \text{ as }s \rightarrow -\infty. \] In particular, \(\varphi(s)\) has an infinite number of zeroes on \((-\infty,0]\).
Let \(p>4\) be an integer. For every \(k>0\) there exists a positive \(C^\infty\)-solution \(\varphi\) defined on all of \(\mathbb R\) having the following properties:
(iii) \(\varphi(s) = ks^{(-3/4 +1/p)} \exp(- \frac{2s^{3/2}}{3\sqrt{3}})(1+O(s^{-3/2}))\) as \(s\rightarrow \infty\), and
(iv) there exist real numbers \(a, b, c\) with \(a \neq 0\), such that \(\varphi(s)=a(-s)^{-3/p}+ (-s)^{(-3/4+1/p)}\times \{b\cos(2(-s)^{3/2}/3\sqrt{3})+c\sin(2(-s)^{3/2}/3\sqrt{3})\}+O(|s|^{-(2/p+3/2)})\) as \(s\rightarrow -\infty\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
35B40 Asymptotic behavior of solutions to PDEs
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