%&amslatex \documentclass{amsart} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{hyp}[thm]{H.} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\MR}{{\mathbb R}} \newcommand{\MN}{{\mathbb N}} \newcommand{\MZ}{{\mathbb Z}} \newcommand{\MC}{{\mathbb C}} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\nn}{\nonumber} %\newcommand{\bea}{\begin{eqnarray}} %\newcommand{\eea}{\end{eqnarray}} %\newcommand{\bay}{\begin{array}} %\newcommand{\eay}{\end{array}} \newcommand{\DS}{\displaystyle} \newcommand{\ol}{\overline} \newcommand{\ul}{\underline} \newcommand{\bs}{\backslash} \newcommand{\hr}{{\frak H}} \newcommand{\db}{{\frak D}} \newcommand{\BO}{\frak B} \newcommand{\sgn}{\text{\rm sgn}} \newcommand{\Tr}{\text{\rm Tr}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\Norm}[1]{\bigl\lVert#1\bigr\rVert} \DeclareMathOperator*{\slim}{s-\lim} \DeclareMathOperator{\Ran}{Ran} %%%%%%% specific Abbreviatons %%%%%%%%%%% \newcommand{\EM}{{\mathcal M}} \newcommand{\LamN}{\Lambda_N(t,x,y)} \newcommand{\Lami}{\Lambda_\infty(t,x,y)} \newcommand{\LamM}{\Lambda_{\mathcal M}(t,x,y)} \newcommand{\pLN}{\partial_x\LamN} \newcommand{\pLi}{\partial_x\Lami} \newcommand{\pLM}{\partial_x\LamM} \newcommand{\hLN}{\hat{\Lambda}_N(t,x,y)} \newcommand{\hLi}{\hat{\Lambda}_\infty(t,x,y)} \newcommand{\hLM}{\hat{\Lambda}_{\mathcal M}(t,x,y)} \newcommand{\limN}{{N\rightarrow\infty}} \newcommand{\aslimN}{\xrightarrow[\limN]{}} \newcommand{\mixpart}{\partial_x^\alpha \partial_y^\beta \partial_t^\gamma} \newcommand{\pa}{\partial^a} \newcommand{\MNi}{\MN\cup\{\infty\}} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \def\theequation{\thesection.\@arabic\c@equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[New soliton solutions for the (m)KP-equation]% {A new class of soliton solutions for the (modified) Kadomtsev-Petviashvili equation} \author{W.~Renger} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{walter@mumathnx4.math.missouri.edu} \keywords{} % Math Subject Classifications \subjclass{} \maketitle %\newcounter{me} \begin{abstract} We construct solutions of the Kadomtsev-Petviashvili equation and its counterpart, the modified Kadomtsev-Petviashvili equation, with an infinite number of solitons by a careful examination of the limits of $N$-soliton solutions as $\limN$. We give sufficient conditions to ensure that these limits exist and satisfy the (modified) Kadomtsev-Petviashvili equation. \end{abstract} \section{Introduction} \setcounter{equation}{0}\label{1} In this note we study solutions of the Kadomtsev-Petviashvili (KP) equation \cite{KP} and the modified Kadomtsev-Petviashvili (mKP) equation \cite{DKJM},\cite{Kon1},\cite{KonDub} with an infinite number of solitons. We start with the well-known $N$-soliton solutions (cf. \cite{AN}--\cite{DKJM},\cite{Dickey}--\cite{GHSS},% \cite{GSchw},\cite{GU},% \cite{Kon1}--\cite{KupWil},\cite{MST},\cite{MZBIM},\cite{MatSal},% \cite{OeRog}--\cite{OW},% \cite{Poe},\cite{PS}) and construct the limits of these solutions for $\limN$. After briefly summarizing the $N$-soliton solutions for the (m)KP-equation in section \ref{2} we will prove under appropriate conditions that the limits of these $N$-soliton solutions for $\limN$ exist. Furthermore, we prove that they actually are solutions of the (m)KP-equation. In section \ref{3} this is done for the KP-equation, in section \ref{4} for the mKP-equation. We mention that similar results are already known for other soliton equations, namely the KdV-equation \cite{GKZduke},\cite{GKZbull}, the Toda-Lattice \cite{GR}; see also \cite{DeSh},\cite{Kamv},\cite{Lun},% \cite{LunMar}, \cite{Mar},\cite{Nov},\cite{Sha}. To the best of our knowledge, the current investigation is the first of this kind for 2+1 dimensional completely integrable evolution equations. \section{$N$-soliton solutions} \setcounter{equation}{0}\label{2} The KP equation $$V_t-6 V V_x + V_{xxx} +3\int_\infty^x dx' V_{yy} = 0 \label{101}$$ has $N$-soliton solutions of the form (cf., e.g., \cite{GH},\cite{GHSS}) $$V=V_N+\lambda\, ,\label{102}$$ where \begin{align} &V_N(t,x,y) = -2\partial_x^2 \ln\det[1+\LamN]\,,\label{103}\\ &\LamN =\Bigl\{c_j(t,y)c_l(t,y)\frac{\exp[-(p_j+q_l)x]}{p_j+q_l} \Bigr\}_{j,l=1}^N\,, \label{104}\\ &c_j(t,y)=c_j\exp\bigl[\tfrac\eps 2(p_j^2-q_j^2)y +\bigl(2(p_j^3+q_j^3)-3\lambda(p_j+q_j)\bigr)t\bigr],\label{105}\\ & \text{for all } (t,x,y)\in\MR^3,\ p_j,q_j>0, \ c_j\in\MR\bs\{0\},\ j=1,\ldots,N, \nn\\&\lambda\in\MR,\ \eps=\pm 1. \nn\end{align} The $N$-soliton solutions $\phi_N$ of the mKP equation $$\phi_t-6\phi^2\phi_x+\phi_{xxx}+3\int_\infty^x dx' \phi_{yy} +6\eps \phi_x\int_\infty^x dx'\phi_y =0\,, \qquad \eps=\pm 1 \label{106}$$ are given by (cf. \cite{GHSS}) $$\phi_N(t,x,y)=-\kappa+\partial_x \ln\Bigl\{\frac{\det[1+\hLN]}{\det[1+\LamN]}\Bigr\}\,, \label{107}$$ with \begin{align}\label{108} &\hLN=\Bigl\{\frac{\kappa-p_j}{\kappa+q_j} c_j(t,y)c_l(t,y) \frac{\exp[-(p_j+q_l)x]}{p_j+q_l} \Bigr\}_{j,l=1}^N\,, \\ \nn &\kappa^2=\lambda, \qquad (t,x,y)\in\MR^3\,, \end{align} and the other quantities defined as in \ref{105}. (The Matrix $\hLN$ is slightly different from the one used in \cite{GHSS}, but the two are related by a similarity transformation, so the corresponding expressions for $\phi_N(t,x,y)$ are actually identical.) The matrices $\LamN$ and $\hLN$ act as operators in $\MC^N$. However, it will be more convenient to embed $\MC^N$ in the Hilbert space of square summable sequences, $\ell^2(\MN)$, in the canonical way. We extend $\LamN$ to an operator in this space by setting all other components equal to zero. {\em We consider both $\LamN$ and $\hLN$ as operators on $\ell^2(\MN)$. All entries, except for the explicitly given first $N\times N$ ones, are defined to be zero.} In the same manner, the operator $1$ will always be understood as the identity operator on $\ell^2(\MN)$. With this convention one obtains \begin{align} \hLN=&D_\infty\LamN\label{109}\,,\\ \intertext{where} D_\infty =&\Bigl\{\frac{\kappa-p_j}{\kappa+q_j}\delta_{j,l}\Big\}_{j,l\in\MN}\,. \label{110}\end{align} ($\delta_{j,l}=1$ for $j=l$ and 0 otherwise.) For later reference we introduce the quantities \begin{align}\label{111} A_N(t,x,y)=&\partial_x\ln\det[1+\LamN] \\ \nn =&\Tr\bigl[[1+\LamN]^{-1}\partial_x\LamN\bigr]\,, \\ \label{112} B_N(t,x,y)=&\partial_x\ln\det[1+\hLN] \\ \nn =&\Tr\bigl[[1+\hLN]^{-1}\partial_x\hLN\bigr]\,. \end{align} Thus we can write \begin{align} V_N(t,x,y)=&-2\partial_x A_N(t,x,y)\,, \label{113}\\ \phi_N(t,x,y)=&-\kappa+B_N(t,x,y)-A_N(t,x,y)\,. \label{114}\end{align} \section{Soliton Limits for the KP-equation} \setcounter{equation}{0}\label{3} The strategy we want to employ is the following: We show that $\lim_\limN V_N(t,x,y)$ is well defined and finite, denote it by $V_\infty(t,x,y)$. Simultaneously, we will prove that the derivatives of this limit are given by the limit of the derivatives of $V_N(t,x,y)$ and that we can also interchange limit and integration. Then we can conclude that the new function $V=V_\infty(t,x,y)+\lambda$ is indeed a solution of the KP-equation. To make this argument work we will have to impose some restrictions on the sequences $\{p_j\}_{j\in\MN}$, $\{q_j\}_{j\in\MN}$, and $\{c_j\}_{j\in\MN}$, by which $V_\infty(t,x,y)$ is defined (see (\ref{103})--(\ref{105})). First of all we note that the $N$-soliton solutions described by (\ref{102})--(\ref{105}) are not necessarily regular, they can have singularities. A convergence argument will be impossible unless we exclude such cases (compare \cite{GHSS}). In order to keep the limit bounded, we will also need boundedness of the sequences $\{p_j\}_{j\in\MN}$, $\{q_j\}_{j\in\MN}$. The condition $\{c_j\}_{j\in\MN}$ has to fulfill is not so clear. A natural condition would be to assume that the trace norm of $\LamN$ is bounded independently of $N$, because then we can define the Fredholm-determinant $\det_1[1+\LamN]$ in the limit $\limN$. However, in general, $\LamN$ is not self-adjoint and thus certainly not a positive operator; there seems to be no simple way to calculate its trace norm. It turns out that it is sufficient to assume (\ref{b4}) given below, which implies boundedness of the trace and of the Hilbert-Schmidt norm of $\LamN$ with respect to $N$. \begin{hyp}\label{h1} The sequences $\{p_j\}_{j\in\MN}$, $\{q_j\}_{j\in\MN}$ are bounded, $00$, and satisfy $$\label{c4} (p_j-p_l)(q_j-q_l)\geq 0,\qquad \text{for all j,l\in\MN}.$$ The sequence $\{c_j\}_{j\in\MN}$, $c_j\in\MR\bs\{0\}$, $j\in\MN$, satisfies $$\label{b4} \sum_{j\in\MN} \frac{c_j^2}{p_j}<\infty, \qquad\qquad \sum_{j\in\MN} \frac{c_j^2}{q_j}<\infty.$$ \end{hyp} We define $$\Lami=\Bigl\{c_j(t,y)c_l(t,y)\frac{\exp[-(p_j+q_l)x]}{p_j+q_l} \Bigr\}_{j,l\in\MN}\,. \label{201}$$ We want to show convergence of $A_N(t,x,y)$ as defined in (\ref{111}) and of its derivatives. To achieve this, we first want to estimate the trace norm $\norm{\pLN}_1$ of the matrix $\pLN$. For later reference we also include a similar estimate on the Hilbert-Schmidt norm $\norm{\LamN}_2$ of the matrix $\LamN$ itself. \begin{lem}\label{21} Assume H.\ref{h1}. Then for all $\EM\in\MNi$, the matrix $\pLM$ is trace class with trace norm bounded by $$\norm{\pLM}_1\leq \sup_{j,l\in\MN}\exp[-(v_j+w_l)x] \sum_{j=1}^\infty c_j(t,y)^2 \exp(-2u_j x)<\infty\,, \label{202}$$ where $u_j=\min(p_j,q_j)$, $v_j=p_j-u_j$, $w_j=q_j-u_j$ for all $j\in\MN$. $\LamM$ is Hilbert-Schmidt with Hilbert-Schmidt norm bounded by $$\label{b30} \norm{\LamM}_2\leq \Bigl\{\sum_{j=1}^{\EM} \frac{c_j(t,y)^2}{2p_j}\exp(-2u_jx) \sum_{l=1}^{\EM} \frac{c_l(t,y)^2}{2q_l}\exp(-2q_lx)\Bigr\}^{\frac12}<\infty\,.$$ \end{lem} \begin{proof} By definition (cf. (\ref{104}) resp. (\ref{201})), \begin{align} &\pLM = -\bigl\{c_j(t,y)c_l(t,y)\exp[-(p_j+q_l)x]\bigr\}_{j,l=1}^\EM \label{205}\\ &\quad=-\bigl\{\exp(-v_jx)\delta_{j,k})\bigr\}_{j,k=1}^\infty \bigl\{c_k(t,y)c_m(t,y)\exp[-(u_k+u_m)x]\bigr\}_{k,m=1}^\EM \nn\\ &\qquad\qquad\times\bigl\{\exp(-w_l x)\delta_{m,l}\bigr\}_{m,l=1}^\infty\,. \nn\end{align} Here the matrix $$\bigl\{c_k(t,y)c_m(t,y)\exp[-(u_k+u_m)x]\bigr\}_{k,m=1}^\EM=:C_\EM(t,x,y) \label{206}$$ is positive, i.e., for all $\varphi\in\ell^2(\MN)$, $(t,x,y)\in\MR^3$, $$\label{a4} (\varphi,C_\EM(t,x,y)\varphi) =\biggl|\sum_{j=1}^{\EM}c_j(t,y)\exp(-u_jx)\varphi_j\biggr|^2\geq 0.$$ Thus its trace norm is simply the trace $$\norm{C_\EM(t,x,y)}_1=\sum_{j=1}^\EM c_j(t,y)^2 \exp(-2u_j x) \leq \sum_{j=1}^\infty c_j(t,y)^2 \exp(-2u_j x)\,. \label{207}$$ Therefore, \begin{align}\label{208} &\norm{\pLM}_1 \leq\norm{\bigl\{\exp(-v_jx)\delta_{j,k})\bigr\}_{j,k=1}^\infty} \\ \nn &\qquad\qquad\times\norm{C_\EM(t,x,y)}_1 \norm{\bigl\{\exp(-w_l x)\delta_{m,l}\bigr\}_{m,l=1}^\infty} \end{align} proves (\ref{202}). (\ref{b30}) follows from a straightforward calculation: \begin{align}\label{b6} &\norm{\LamM}_2^2=\Tr\bigl[\LamM^*\LamM\bigr] \\ \nn &\quad=\sum_{j,l=1}^{\EM} \frac{c_j(t,y)^2 c_l(t,y)^2}{(p_j+q_l)^2}\exp[-2(p_j+q_l)x] \\ \nn &\quad\leq \sum_{j=1}^{\EM} \frac{c_j(t,y)^2}{2p_j}\exp(-2p_jx) \sum_{l=1}^{\EM} \frac{c_l(t,y)^2}{2q_l}\exp(-2q_lx)<\infty \end{align} by (\ref{b4}). \end{proof} We can also extend this lemma to higher order derivatives. \begin{lem}\label{2a} Suppose H.\ref{h1} holds. For all $\alpha,\beta,\gamma \in\MN_0$, let $a$ be the multi-index $a=(\alpha,\beta,\gamma)$, $\pa=\mixpart$. Then for all $\EM\in\MNi$, \begin{align}\label{203} &\norm{\pa\pLM}_1\leq const.(a)\norm{\pLM}_1\leq const.(a,t,x,y), \end{align} $$\label{b1} \norm{\pa\LamM}_2\leq const.(a)\norm{\LamM}_2\leq const.(a,t,x,y),$$ with the constants independent of $\EM$ and locally uniform with respect to $(t,x,y)\in\MR^3$ (i.e., uniform in all compact subsets of $\MR^3$). Furthermore, $$\Norm{\pa\pLi-\pa\pLN}_1\aslimN 0, \label{204}$$ $$\label{b2} \Norm{\pa\Lami-\pa\LamN}_2\aslimN 0$$ locally uniformly with respect to $(t,x,y)\in\MR^3$. \end{lem} \begin{proof} To prove (\ref{203}) we use an induction argument: Since the estimate (\ref{202}) is independent of $\EM$ and locally uniform with respect to $(t,x,y)\in\MR^3$, we have already proved it for $\alpha=\beta=\gamma=0$. Assume it holds for $\alpha,\beta,\gamma\in\MN_0$. Then \begin{align}\label{209} &\norm{\partial_x\pa\pLM}_1= \Norm{\bigl\{-(p_j+q_l)\{\pa\pLM\}_{j,l}\bigr\}_{j,l=1}^\EM}_1 \\ \nn &\quad\leq\norm{\bigl\{p_j\delta_{j,k}\bigr\}_{j,k=1}^\infty} \norm{\pa\pLM}_1 \\ \nn &\qquad+\norm{\pa\pLM}_1 \norm{\bigl\{q_l\delta_{m,l}\bigr\}_{m,l=1}^\infty} \\ \nn &\quad\leq const.\norm{\pa\pLM}_1 \end{align} proves (\ref{203}) for $\alpha+1,\beta,\gamma$. The induction steps $\beta\rightarrow\beta+1$ and $\gamma\rightarrow\gamma+1$ are done in a similar manner. There we get more complicated factors, but we can still expand them to get an expression analogous to (\ref{209}). (\ref{b1}) is proved by exactly the same induction argument, starting from (\ref{b30}) for $\alpha=\beta=\gamma=0$. Now let $P_N$ be the projection from $\ell^2(\MN)$ to $\MC^N$, $Q_N=1-P_N$. Then for all positive self-adjoint trace class operators $F$ on $\ell^2(\MN)$, \begin{align}\label{210} &\norm{F-P_N F P_N}_1 = \norm{Q_N F +P_N F Q_N}_1 \leq 2\norm{Q_N F}_1 \\ \nn &\quad\leq 2\norm{Q_N F^{\frac12}}_2\norm{F^{\frac12}}_2 =2 \norm{Q_N F Q_N}_1^{\frac12}\norm{F}_1^{\frac12} \aslimN 0\,. \end{align} The matrices $\pLi$ and $Q_N\pLi Q_N$ are both positive, so by (\ref{202}) we can use (\ref{210}) to get \begin{align}\label{211} &\norm{\pLi-\pLN}_1 \aslimN 0\,. \end{align} Because the bound in (\ref{203}) is locally uniform with respect to $(t,x,y)\in\MR^3$, this convergence is locally uniform, too. So we have (\ref{204}) for $\alpha=\beta=\gamma=0$ and can use an induction argument as in the proof of (\ref{203}) to infer the statement for general $\alpha$, $\beta$, and $\gamma$. The proof of (\ref{b2}) is similar. For $\alpha=\beta=\gamma=0$, \begin{align}\label{b7} &\norm{\Lami-\LamN}_2^2 \\ \nn &\quad=\Tr\bigl[\bigl(Q_N\Lami+P_N\Lami Q_N\bigr)^* \bigl(Q_N\Lami+P_N\Lami Q_N\bigr)\bigr] \\ \nn &\quad=\Tr\bigl[\Lami Q_N\Lami+Q_N\Lami P_N\Lami Q_N\bigr] \\ \nn &\quad \leq 2\sum_{j=1}^\infty\sum_{l=N+1}^\infty c_j(t,y)^2c_l(t,y)^2\frac{\exp[-2(p_j+q_l)x]}{(p_j+q_l)^2} \aslimN 0 \end{align} by (\ref{b4}) locally uniformly with respect to $(t,x,y)\in\MR^3$. Induction in $\alpha$, $\beta$, and $\gamma$ completes the proof. \end{proof} Getting $[1+\LamN]^{-1}$ under control turns out to be a bit more involved. We find \begin{lem}\label{22} Suppose H.\ref{h1} holds. Then for all $\EM\in\MNi$, $$\Norm{[1+\LamM]^{-1}}\leq const.(t,x,y)\,, \label{212}$$ with the constant independent of $\EM$ and locally uniform with respect to $(t,x,y)\in\MR^3$. Furthermore, $$\label{214} \Norm{[1+\Lami]^{-1}-[1+\LamN]^{-1}}_2\aslimN 0$$ locally uniformly with respect to $(t,x,y)\in\MR^3$. \end{lem} \begin{proof} If we consider $\LamN$ as an operator on $\MC^N$, we can calculate its determinant (see Polya, Szeg\"o \cite{PoSz}, p.92) \begin{align}\label{215} &\det\LamN=\exp\Bigl[-\sum_{j=1}^N(p_j+q_j)x\Bigr]\prod_{j=1}^N c_j^2\, \Bigl[\prod_{r,s=1}^N(p_r+q_s)\Bigr]^{-1} \\ \nn &\qquad\qquad\times \prod_{\genfrac{}{}{0pt}{1}{l,k=1}{k>l}}^{N}(p_l-p_k)(q_l-q_k) \geq 0 \nn\end{align} by (\ref{c4}). By expanding, we find \begin{align}\label{216} &\det[1+\LamN] = 1+\det[\LamN]+\sum_{j_1=1}^N \det[\LamN^{j_1}] \\ \nn &\qquad +\hspace{-2ex} \sum_{\genfrac{}{}{0pt}{1}{j_1,j_2=1}{j_10 \end{align} by (\ref{217}) and (\ref{b4}). But by (\ref{b2}) and Theorem 9.2 of \cite{Simon} this implies $$\label{b10} \det\nolimits_2[1+\Lami]=\lim_\limN\det\nolimits_2[1+\LamN] \geq const.(t,x,y)>0.$$ Again by \cite{Simon}, Theorem 9.2, this guarantees that $1+\Lami$ is invertible. For $(t,x,y)\in\MR^3$ fixed, $$\label{b11} \norm{[1+\Lami]^{-1}}= c(t,x,y)<\infty,$$ with some constant $c(t,x,y)$. If $(t,x,y)$ and $(\tau,\xi,\eta)$ are in $\MR^3$, then we can calculate the Hilbert-Schmidt norm \begin{align}\label{c1} &\norm{\Lami-\Lambda_\infty(\tau,\xi,\eta)}_2^2 \\ \nn &\quad=\sum_{j,l=1}^{\EM} \Bigl|\frac{c_j(t,y) c_l(t,y)}{p_j+q_l}\exp[-(p_j+q_l)x] -\frac{c_j(\tau,\eta) c_l(\tau,\eta)}{p_j+q_l}\exp[-(p_j+q_l)\xi]\Bigr|^2 \\ \nn &\quad\xrightarrow[(\tau,\xi,\eta)\rightarrow (t,x,y)]{} 0 \end{align} by the Weierstrass test. (The sum converges by (\ref{b4}) and in the limit $(\tau,\xi,\eta)\rightarrow (t,x,y)$ each term converges to 0.) Therefore there is an $\eps>0$ such that for all $(\tau,\xi,\eta)$ with $|(\tau,\xi,\eta)-(t,x,y)|<\eps$, $\norm{\Lami-\Lambda_\infty(\tau,\xi,\eta)}_2<\frac{1}{2c(t,x,y)}\,$, which implies \begin{align}\label{c2} &\Norm{[1+\Lambda_\infty(\tau,\xi,\eta)]^{-1}}\leq \Norm{[1+\Lami]^{-1}} \\ \nn &\qquad\quad\times\Norm{\bigl\{1+[1+\Lami]^{-1} [\Lambda_\infty(\tau,\xi,\eta)-\Lami]\bigr\}^{-1}} \\ \nn &\quad\leq c(t,x,y)\Bigl\{1-\Norm{[1+\Lami]^{-1}}\, \Norm{[\Lambda_\infty(\tau,\xi,\eta)-\Lami]}\Bigr\}^{-1} \\ \nn &\quad\leq 2 c(t,x, y). \end{align} If $K\subset\MR^3$ is compact, we can cover it with a finite number of such $\eps$-neighborhoods and $c_K:=\sup_{(t,x,y)\in K}[c(t,x,y)]$ is finite. By (\ref{b2}) there is an $N_K\in\MN$ such that for all $\EM\in\MNi$, $\EM\geq N_K$, and for $(t,x,y)\in K$, $\norm{\Lami-\LamM}\leq\frac{1}{2c_K}$. Then for such $\EM\geq N_K$ and $(t,x,y)\in K$, \begin{align}\label{b13} &\Norm{[1+\LamM]^{-1}}\leq 2c_K \end{align} by an estimate analogous to (\ref{c2}). Together with (\ref{b12}) this proves (\ref{212}). (\ref{214}) is an immediate consequence of (\ref{212}) and (\ref{b2}), since \begin{align} &\Norm{[1+\Lami]^{-1}-[1+\LamN]^{-1}}_2 \\ \nn &\quad \leq \Norm{[1+\Lami]^{-1}} \Norm{[\Lami-\LamN]}_2 \Norm{[1+\LamN]^{-1}} \\ \nn &\quad\aslimN 0 \end{align} locally uniformly with respect to $(t,x,y)\in\MR^3$. \end{proof} \par From these three lemmas we can conclude the convergence of $A_N(t,x,y)$ in the limit $\limN$. \begin{lem}\label{23} Suppose H.\ref{h1} holds and define $$\label{221} A_\infty(t,x,y)=\Tr\bigl[[1+\Lami]^{-1}\partial_x\Lami\bigr]\,.$$ For all $\alpha,\beta,\gamma\in\MN_0$, let $\pa=\mixpart$. Then $$\label{222} \pa A_\infty(t,x,y)=\lim_\limN\pa A_N(t,x,y)\,,$$ the convergence being locally uniform with respect to $(t,x,y)\in\MR^3$. \end{lem} \begin{proof} (\ref{221}) is well defined because of Lemmas \ref{21} and \ref{22}. Convergence of $A_N(t,x,y)\aslimN A_\infty(t,x,y)$ then follows from \begin{align}\label{223} & \abs{A_\infty(t,x,y)-A_N(t,x,y)} \\ \nn &\quad \leq \Norm{[1+\Lami]^{-1} -[1+\LamN]^{-1}}\,\norm{\pLi}_1 \\ \nn & \qquad+\norm{[1+\LamN]^{-1}}\,\Norm{[\pLi-\pLN]}_1\aslimN 0 \end{align} by (\ref{202}), (\ref{204}), (\ref{212}), (\ref{214}). All estimates are locally uniform with respect to $(t,x,y)\in\MR^3$, so we have (\ref{222}) for $\alpha=\beta=\gamma=0$. To prove the convergence for general multi-indices $a=(\alpha,\beta,\gamma)$ we note that $\pa A_N(t,x,y)$ is of the form $$\label{224} \pa A_N(t,x,y) =\sum_{k=1}^{n(a)}\Tr[K_{N,k}^a(t,x,y)],$$ with $n(a)\in\MN$ and matrices \begin{align}\label{225} &K_{N,k}^{a}(t,x,y) =\pm[1+\LamN]^{-1}\partial^{a_1}\LamN [1+\LamN]^{-1} \\ \nn &\qquad\times\ \ldots\ \times [1+\LamN]^{-1}\partial^{a_{s(k)}}\partial_x\LamN \end{align} for some $s(k)\in\MN_0$ and multi-indices $a_1,\,\ldots,a_{s(k)}$, $a_1+\ldots+a_{s(k)}=a$. By Lemmas \ref{2a} and \ref{22} $\norm{K_{N,k}^{a}(t,x,y)}_1\leq const.(a,k,t,x,y)$ with the constant locally uniform with respect to $(t,x,y)\in\MR^3$ and independent of $N$. Thus $$\label{226} K_{\infty,k}^a(t,x,y):=\lim_\limN K_{N,k}^{a}(t,x,y)$$ exists and is trace class. Then \begin{align}\label{227} &\bigl|\Tr[K_{\infty,k}^a(t,x,y)] -\Tr[K_{N,k}^{a}(t,x,y)]\bigr| \\ \nn &\quad \leq \bigl|\Tr\bigl[ \bigl\{[1+\Lami]^{-1}-[1+\LamN]^{-1}\bigr\} \partial^{a_1}\Lami \\ \nn &\qquad\qquad\times[1+\Lami]^{-1}\times\ \ldots\ \times [1+\Lami]^{-1}\partial^{a_{s(k)}}\partial_x\Lami \bigr]\bigr| \\ \nn &\qquad+ \bigl|\Tr\bigl[[1+\LamN]^{-1}\bigl\{\partial^{a_1}\Lami- \partial^{a_1}\LamN\bigr\} [1+\Lami]^{-1} \\ \nn &\qquad\qquad\times\ \ldots\ \times [1+\Lami]^{-1}\partial^{a_{s(k)}}\partial_x\Lami\bigr]\bigr| \\ \nn &\qquad + \\ \nn &\qquad\ \, \vdots \\ \nn &\qquad+\bigl|\Tr\bigl[[1+\LamN]^{-1}\partial^{a_1}\LamN [1+\LamN]^{-1} \\ \nn &\qquad\qquad\times\ \ldots\ \times [1+\LamN]^{-1}\bigl\{\partial^{a_{s(k)}}\pLi-\partial^{a_{s(k)}}\pLN \bigr\}\bigr]\bigr| \\ \nn &\quad\leq \Norm{[1+\Lami]^{-1}-[1+\LamN]^{-1}} \\ \nn &\qquad\quad\times \Norm{ \partial^{a_1}\Lami [1+\Lami]^{-1} \\ \nn &\qquad\qquad\times\ \ldots\ \times [1+\Lami]^{-1}\partial^{a_{s(k)}}\partial_x\Lami}_1 \\ \nn &\qquad+\Norm{[1+\LamN]^{-1}}\Norm{\partial^{a_1}\Lami- \partial^{a_1}\LamN}_2 \\ \nn &\qquad\quad\times\Norm{ [1+\Lami]^{-1} \times\ \ldots\ \times [1+\Lami]^{-1}\partial^{a_{s(k)}}\partial_x\Lami}_1 \end{align} \begin{align}\nn &\qquad + \\ \nn &\qquad\ \, \vdots \\ \nn &\qquad+\Norm{[1+\LamN]^{-1}\partial^{a_1}\LamN [1+\LamN]^{-1} \\ \nn &\qquad\qquad\times\ \ldots\ \times [1+\LamN]^{-1}} \\ \nn &\qquad\quad\times\Norm{\partial^{a_{s(k)}}\pLi-\partial^{a_{s(k)}}\pLN}_1 \aslimN 0 \end{align} locally uniformly with respect to $(t,x,y)\in\MR^3$ by (\ref{203})--(\ref{b2}), (\ref{212}), and (\ref{214}). \end{proof} Now we are prepared to prove our first main result. \begin{thm}\label{24} Assume H.\ref{h1}. Then the function $$\label{228} V(t,x,y)=V_\infty(t,x,y)+\lambda:=-2\partial_x A_\infty(t,x,y)+\lambda$$ is a solution of the KP-equation (\ref{101}). \end{thm} \begin{proof} Applying the Weierstrass test to (\ref{b30}) shows that, independently of $\EM\in\MNi$, $\norm{\LamM}_2\xrightarrow[x\rightarrow+\infty]{}0$. Thus for large enough $x$, $$\label{b34} \norm{[1+\LamM]^{-1}}\leq\bigl\{1-\norm{\LamM}\bigr\}^{-1} \xrightarrow[x\rightarrow+\infty]{}1.$$ In a similar fashion by (\ref{202}) $\norm{\pLM}_1\xrightarrow[x\rightarrow+\infty]{}0$, by (\ref{203}) the same holds for its derivatives, so $\partial_y^2 A_\EM(t,x,y)\xrightarrow[x\rightarrow+\infty]{}0$ for all $\EM\in\MNi$. Boundedness of all derivatives guarantees that we can interchange the order of differentiation, so \begin{align}\label{231} &\int_\infty^x dx' \partial^2_y V_\infty(t,x',y) =-2\int_\infty^x dx'\partial_x\partial_y^2 A_\infty(t,x',y) =-2\partial_y^2 A_\infty(t,x,y) \\ \nn &\quad=\lim_\limN(-2\partial_y^2 A_N(t,x,y)) =\lim_\limN \int_\infty^x dx' \partial^2_y V_N(t,x',y). \end{align} By Lemma \ref{23} we can also interchange the limits $\limN$ with derivatives, thus we have similar convergence results for all the other terms in (\ref{101}) and the theorem follows from the fact that $V=V_N(t,x,y)+\lambda$ solves the KP-equation. \end{proof} \begin{rem} Presumably, our hypotheses are not optimal; however, they are general enough to cover a large class of solutions. In particular, we remark that the only restriction on the sequences $\{p_j\}_{j\in\MN}$, $\{q_j\}_{j\in\MN}$ is the boundedness condition and the requirement of having the $N$-soliton solutions regular. The only condition which is technical and could perhaps be weakened is the one imposed on the sequence $\{c_j\}_{j\in\MN}\,$. \end{rem} \section{Soliton Limits for the mKP-equation} \setcounter{equation}{0}\label{4} We follow the same steps we used for the KP-equation; we start with the $N$-soliton solutions given by (\ref{107}). H.\ref{h1} is not sufficient to ensure that these solutions are regular (cf. Rmk. \ref{46}). We need the additional hypothesis (cf. (\ref{108})) \begin{hyp}\label{h4} Suppose that either $\kappa=\lambda^{\frac12}\geq\sup_{j\in\MN}(p_j)$ or $\kappa<\inf_{j\in\MN}(-q_j)$. \end{hyp} Now we can extend the lemmas from the previous section \ref{3} to $\hLN$, respectively, $B_N(t,x,y)$. \begin{lem}\label{32} Assume H.\ref{h1} and H.\ref{h4}. Let $a$ be a multi-index $a=(\alpha,\beta,\gamma)$, $\alpha,\beta,\gamma\in\MN_0$ arbitrary, and abbreviate $\pa=\mixpart$. Then for all $\EM\in\MNi$, $$\label{301} \norm{\pa\partial_x\hLM}_1 \leq const.(a)\norm{\pLM}_1\leq const.(a,t,x,y),$$ $$\label{b20} \norm{\pa\hLM}_2\leq const.(a)\norm{\LamM}_2 \leq const.(a,t,x,y),$$ with the constants independent of $\EM$ and locally uniform with respect to $(t,x,y)\in\MR^3$. Moreover, $$\label{302} \Norm{\pa\partial_x\hLi-\pa\partial_x\hLN}_1\aslimN 0,$$ $$\label{b21} \Norm{\pa\hLi-\pa\hLN}_2\aslimN 0$$ locally uniformly with respect to $(t,x,y)\in\MR^3$. Furthermore, $$\label{303} \Norm{[1+\hLM]^{-1}}\leq const.(t,x,y),$$ with the constant locally uniform with respect to $(t,x,y)\in\MR^3$, and $$\label{304} \Norm{[1+\hLi]^{-1}-[1+\hLN]^{-1}}_2\aslimN 0$$ locally uniformly with respect to $(t,x,y)\in\MR^3$. \end{lem} \begin{proof} (\ref{301}) -- (\ref{b21}) follow from Lemma \ref{2a} and (\ref{109}). (The matrix $D_\infty(t,x,y)$ in (\ref{109}) is independent of $t,x,y$ and bounded by H.\ref{h1} and H.\ref{h4}.) In order to prove (\ref{303}), we note that we can rewrite (\ref{109}) as $\hLN=D_N \LamN$, where $D_N$, the restriction of $D_\infty$ to $\MC^N$, is an $N\times N$ matrix. Then $$\label{305} \det\hLN=\prod_{j=1}^N\frac{\kappa-p_j}{\kappa+q_j}\det\LamN \geq 0$$ by H.\ref{h4} and (\ref{215}). Thus we can follow the proof of Lemma \ref{22} to infer (\ref{303}) and (\ref{304}). \end{proof} \begin{lem}\label{43} Suppose H.\ref{h1} and H.\ref{h4} hold. Define $$\label{306} B_\infty(t,x,y)=\Tr\bigl[[1+\hLi]^{-1}\partial_x\hLi\bigr].$$ For all $\alpha,\beta,\gamma\in\MN_0$, let $\pa=\mixpart$. Then $$\label{307} \pa B_N(t,x,y)\aslimN\pa B_\infty(t,x,y)$$ locally uniformly with respect to $(t,x,y)\in\MR^3$. \end{lem} \begin{proof} Because of the boundedness of $D_\infty$ (cf. (\ref{109}) resp. (\ref{110})) we can follow the proof of Lemma \ref{23} step by step. \end{proof} By Lemmas \ref{23} and \ref{43} $$\label{308} \phi_N(t,x,y)\aslimN\phi_\infty(t,x,y) =-\kappa+B_\infty(t,x,y)-A_\infty(t,x,y)$$ and $\phi_\infty(t,x,y)$ satisfies the differentiated version of the mKP-equation. However, in order to prove that it satisfies the mKP-equation in integral form (\ref{106}) we need a slight strengthening of H.\ref{h1}, namely hypothesis \begin{hyp}\label{h5} The sequence $\{c_j\}_{j\in\MN}$, $c_j\in\MR\bs\{0\}$, $j\in\MN$, satisfies $$\label{309} \sum_{j=1}^\infty \frac{c_j^2}{\min(p_j,q_j)}<\infty.$$ \end{hyp} With this hypothesis we finally obtain \begin{thm}\label{35} Suppose H.\ref{h1}, H.\ref{h4}, and H.\ref{h5} hold. Then the function $$\label{310} \phi_\infty(t,x,y)=-\kappa+B_\infty(t,x,y)-A_\infty(t,x,y)$$ satisfies the mKP-equation (\ref{106}). \end{thm} \begin{proof} We know that $\phi_N(t,x,y)$ solves the mKP-equation for every $N\in\MN$. Lemmas \ref{23} and \ref{43} allow us to exchange the limit $\limN$ with derivatives. So we just have to check whether we can interchange the limit with the integral, i.e., we have to prove $$\label{311} \int_\infty^x dx'\partial_y^\alpha \phi_N(t,x,y) \aslimN \int_\infty^x dx' \partial_y^\alpha \phi_\infty(t,x,y), \qquad \alpha=1,2.$$ For that purpose we want to rely on dominated convergence. The integrand converges and is bounded by \begin{align}\label{312} &\abs{\partial_y^\alpha\phi_N(t,x,y)} \leq\abs{\partial_y^\alpha B_N(t,x,y)}+\abs{\partial_y^\alpha A_N(t,x,y)}, \qquad \alpha=1,2. \end{align} Here \begin{align}\label{c5} &|\partial_y A_N(t,x,y)|=\bigl|\Tr\bigl[ \bigl\{-[1+\LamN]^{-1}\partial_y\LamN [1+\LamN]^{-1} \\ \nn &\qquad\quad+[1+\LamN]^{-1}\partial_y\bigr\}\partial_x\LamN\bigr]\bigr| \\ \nn &\quad\leq\bigl\{\norm{[1+\LamN]^{-1}}\, \norm{\partial_y\LamN}\,\norm{[1+\LamN]^{-1}} \\ \nn &\qquad\quad+\norm{[1+\LamN]^{-1}}const.\bigr\}\norm{\pLN}_1 \\ \nn &\quad\leq const.(t,x,y)\norm{\pLN}_1\,. \end{align} The constant is independent of $N$ and locally uniform with respect to $(t,x,y)\in\MR^3$. Since $\norm{\partial_y\LamN} \xrightarrow[x\rightarrow+\infty]{}0$ (apply the Weierstrass test to (\ref{b30}) and use (\ref{b1})) and $\norm{[1+\LamN]^{-1}}\xrightarrow[x\rightarrow+\infty]{}1$ (by (\ref{b34})), this constant converges in the limit $x\rightarrow+\infty$, so we can choose it uniform for $x$ on a half-axis $[a_0,\infty)$ and $(t,y)$ in a compact subset of $\MR^2$. The same argument can be applied to $\partial_y^2 A_N(t,x,y)$, $\partial_y B_N(t,x,y)$, and $\partial_y^2 B_N(t,x,y)$. Therefore it is sufficient to estimate (using (\ref{202}) and the notation introduced in Lemma \ref{21}), \begin{align}\label{313} &\int_\infty^x dx'\norm{\partial_x\Lambda_\infty(t,x',y)}_1 \\ \nn &\quad\leq \sup_{j,l\in\MN}\exp[-(v_j+w_l)x] \int_\infty^x dx'\sum_{j=1}^\infty c_j^2\exp(-2u_j x') \\ \nn &\quad = \sup_{j,l\in\MN}\exp[-(v_j+w_l)x] \sum_{j=1}^\infty \frac{c_j^2}{2u_j}\exp(-2u_j x)<\infty \end{align} by H.\ref{h5}. (The interchange of the integral with the sum is justified because all terms in the sum are positive.) Thus $\abs{\partial_y\phi_N(t,x,y)}$ and $\abs{\partial_y^2\phi_N(t,x,y)}$ are both bounded by an integrable function and (\ref{311}) follows by the dominated convergence theorem. \end{proof} \begin{rem}\label{46} H.\ref{h4} is necessary in the sense that if there is a $p_{j_0}$ with $p_{j_0}>\kappa\geq 0$ or an $q_{j_0}$ with $-q_{j_0}<\kappa\leq 0$, then for this index $j_0$, $\frac{\kappa-p_{j_0}}{\kappa+q_{j_0}}<0$. By following a procedure similar to the proof of \cite{GHSS}, Prop. 3.2 we can then show that there is a choice for the sequence $\{c_j\}_{j\in\MN}$ such that $\det[1+\hLN]$ changes sign at least once. 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