Analyticity of solitary wave solutions to generalized Kadomtsev-Petviashvili equations (Q2719737)

The authors consider solitary wave solutions \(u(t,x,y)=U(x-ct,y)\) for generalized Kadomtsev-Petviashvili equations in \(\mathbb R^2\) NEWLINE\[NEWLINE u_t+(f(u))_x+g(D_x)u+\varepsilon v_y=0,\quad v_x=u_y\tag{1}NEWLINE\]NEWLINE and \(u(t,x,y,z)=U(x-ct,y,z)\) for the same equations in \(\mathbb R^3\) NEWLINE\[NEWLINE u_t+(f(u))_x+g(D_x)u+a v_y+bw_z=0,\quad v_x=u_y,\quad w_x=u_z. \tag{2}NEWLINE\]NEWLINE The usual Kadomtsev-Petviashvili equations correspond to the case \(g(D_x)u=u_{xxx},\) \(f(u)=u^2\), more generally \(f(u)=u^p,p\in\mathbb R\), \(u\geq 0\). Assumptions:NEWLINENEWLINENEWLINE(I) \(g\in C^2(\mathbb R)\), \(\xi g(\xi)\geq c\xi^4\); NEWLINENEWLINENEWLINE(II) \(\forall R>0\), \(\exists M>0\), \(\forall x:|x|<R\), \(\forall n\in \mathbb N\), \(|f^{(n)}(x)|<M^{n+1}n!\); NEWLINENEWLINENEWLINE(III) \(\exists r\geq 0\), \(\exists m\in \mathbb N\), \(m\geq r\), \(k_1(|\xi|^{2r+1}+|\xi|^{2m+1})\geq |g(\xi)|\geq k_2(|\xi|^{2r+1}+|\xi|^{2m+1});\) NEWLINENEWLINENEWLINE(IV) \(f(x)\sim c_1x^{p_1}\) near \(x=0\), \(f(x)\sim c_2x^{p_2}\) for \(|x|\to\infty\); NEWLINENEWLINENEWLINE(V) for \(F(x)=\int_0^xf(s) ds\) \(\exists H\in C^1(\mathbb R)\), \(\exists \eta:0<\eta<1\), \(\exists C>0\), \(\forall x\in \mathbb R\), \(|H(x)|\leq\eta|F(x)|\) and \(F(x)+H(x)\leq Cxf(x)\). NEWLINENEWLINENEWLINEUnder these assumptions the authors prove the existence of a wave solution \(u\) of (1) and (2) for \(c>0\) in some natural class of decreasing functions, its smoothness and real analyticity. The proof follows the variation path, and the results extend corresponding assertions of \textit{A. de Bouard} and \textit{J.-C. Saut} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 211-236 (1997; Zbl 0883.35103)].