{"id":32793,"date":"2022-09-24T19:15:15","date_gmt":"2022-09-24T11:15:15","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=32793"},"modified":"2026-03-09T14:31:48","modified_gmt":"2026-03-09T06:31:48","slug":"pde032_150603","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%81%8f%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%b0%8e%e8%ab%96-introduction-to-partial-differential-equations-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e6%9d%8e%e6%a6%ae%e8%80%80%e8%80%81\/pde032_150603","title":{"rendered":"Nonlinear Problems (III) &#8211; : Three famous universal nonlinear PDEs &#8211; kdV, s-G, and NLS equations. Completely integrable systems. s-G equation and the travelling wave solutions. NLS equation and the solitary wave solutions."},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"32793\" class=\"elementor elementor-32793\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-16700d17 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"16700d17\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-100 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1832f798\" data-id=\"1832f798\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4a3731c elementor-widget elementor-widget-breadcrumbs\" data-id=\"4a3731c\" data-element_type=\"widget\" data-widget_type=\"breadcrumbs.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<link rel=\"stylesheet\" href=\"https:\/\/ocw.nycu.edu.tw\/wp-content\/plugins\/elementor-pro\/assets\/css\/widget-theme-elements.min.css\"><p id=\"breadcrumbs\"><span><span><a href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-6ad968e6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6ad968e6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-52987d63\" data-id=\"52987d63\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e5808cb elementor-widget elementor-widget-heading\" data-id=\"e5808cb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">Nonlinear Problems (III) - : Three famous universal nonlinear PDEs - kdV, s-G, and NLS equations. Completely integrable systems. s-G equation and the travelling wave solutions. NLS equation and the solitary wave solutions.\n<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7aec5b11 elementor-widget elementor-widget-video\" data-id=\"7aec5b11\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/GFDuhqdmy8s&quot;,&quot;modestbranding&quot;:&quot;yes&quot;,&quot;lazy_load&quot;:&quot;yes&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-video .elementor-widget-container{overflow:hidden;transform:translateZ(0)}.elementor-widget-video .elementor-wrapper{aspect-ratio:var(--video-aspect-ratio)}.elementor-widget-video .elementor-wrapper iframe,.elementor-widget-video .elementor-wrapper video{height:100%;width:100%;display:flex;border:none;background-color:#000}@supports not (aspect-ratio:1\/1){.elementor-widget-video .elementor-wrapper{position:relative;overflow:hidden;height:0;padding-bottom:calc(100% \/ var(--video-aspect-ratio))}.elementor-widget-video .elementor-wrapper iframe,.elementor-widget-video .elementor-wrapper video{position:absolute;top:0;right:0;bottom:0;left:0}}.elementor-widget-video .elementor-open-inline .elementor-custom-embed-image-overlay{position:absolute;top:0;left:0;width:100%;height:100%;background-size:cover;background-position:50%}.elementor-widget-video .elementor-custom-embed-image-overlay{cursor:pointer;text-align:center}.elementor-widget-video .elementor-custom-embed-image-overlay:hover .elementor-custom-embed-play i{opacity:1}.elementor-widget-video .elementor-custom-embed-image-overlay img{display:block;width:100%}.elementor-widget-video .e-hosted-video .elementor-video{-o-object-fit:cover;object-fit:cover}.e-con-inner>.elementor-widget-video,.e-con>.elementor-widget-video{width:var(--container-widget-width);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-inner-section elementor-element elementor-element-7c450277 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7c450277\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-48607404\" data-id=\"48607404\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-652aa46c elementor-widget elementor-widget-penci-button\" data-id=\"652aa46c\" data-element_type=\"widget\" data-widget_type=\"penci-button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pcbtn-wrapper pcbtn-align-left\">\r\n\t\t\t<a class=\"pcbtn pcbtn-icon-left elementor-animation-grow\" href=\"\/?post_type=course_page&#038;p=29373\">\r\n\t\t\t\t<span class=\"pcbtn-wrapperin\">\r\n\t\t\t\t\t<span class=\"pcbtn-content\"><svg class=\"e-font-icon-svg e-fas-arrow-left\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M257.5 445.1l-22.2 22.2c-9.4 9.4-24.6 9.4-33.9 0L7 273c-9.4-9.4-9.4-24.6 0-33.9L201.4 44.7c9.4-9.4 24.6-9.4 33.9 0l22.2 22.2c9.5 9.5 9.3 25-.4 34.3L136.6 216H424c13.3 0 24 10.7 24 24v32c0 13.3-10.7 24-24 24H136.6l120.5 114.8c9.8 9.3 10 24.8.4 34.3z\"><\/path><\/svg>\u8fd4\u56de\u8ab2\u7a0b\u9996\u9801<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-fc8fc3b\" data-id=\"fc8fc3b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-546b36ef elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"546b36ef\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-divider{--divider-border-style:none;--divider-border-width:1px;--divider-color:#0c0d0e;--divider-icon-size:20px;--divider-element-spacing:10px;--divider-pattern-height:24px;--divider-pattern-size:20px;--divider-pattern-url:none;--divider-pattern-repeat:repeat-x}.elementor-widget-divider .elementor-divider{display:flex}.elementor-widget-divider .elementor-divider__text{font-size:15px;line-height:1;max-width:95%}.elementor-widget-divider .elementor-divider__element{margin:0 var(--divider-element-spacing);flex-shrink:0}.elementor-widget-divider .elementor-icon{font-size:var(--divider-icon-size)}.elementor-widget-divider .elementor-divider-separator{display:flex;margin:0;direction:ltr}.elementor-widget-divider--view-line_icon .elementor-divider-separator,.elementor-widget-divider--view-line_text .elementor-divider-separator{align-items:center}.elementor-widget-divider--view-line_icon .elementor-divider-separator:after,.elementor-widget-divider--view-line_icon 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.elementor-divider__element{margin-right:0}.elementor-widget-divider:not(.elementor-widget-divider--view-line_text):not(.elementor-widget-divider--view-line_icon) .elementor-divider-separator{border-top:var(--divider-border-width) var(--divider-border-style) var(--divider-color)}.elementor-widget-divider--separator-type-pattern{--divider-border-style:none}.elementor-widget-divider--separator-type-pattern.elementor-widget-divider--view-line .elementor-divider-separator,.elementor-widget-divider--separator-type-pattern:not(.elementor-widget-divider--view-line) .elementor-divider-separator:after,.elementor-widget-divider--separator-type-pattern:not(.elementor-widget-divider--view-line) .elementor-divider-separator:before,.elementor-widget-divider--separator-type-pattern:not([class*=elementor-widget-divider--view]) .elementor-divider-separator{width:100%;min-height:var(--divider-pattern-height);-webkit-mask-size:var(--divider-pattern-size) 100%;mask-size:var(--divider-pattern-size) 100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-da19150 elementor-widget elementor-widget-shortcode\" data-id=\"da19150\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-66\" class=\"tablepress tablepress-id-66\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\">\u7b2c\u4e00\u9031<\/td><td class=\"column-2\">PDE\u5c0e\u8ad6.<br \/>\nFundamental differences between PDE and ODE.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=29440\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\">\u7b2c\u4e8c\u9031<\/td><td class=\"column-2\">First and second order linear wave equations;<br \/>\nTransport equations<br \/>\nCharacteristic lines;<br \/>\nTravelling wave solutions.<br \/>\nWave equations with dispersion, dissipation, and nonlinearity.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32671\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\">\u7b2c\u4e8c\u9031<\/td><td class=\"column-2\">Classical linear wave equations with travelling wave solutions.<br \/>\nDispersive linear wave equations.<br \/>\nDissipative linear wave equations.<br \/>\nNonlinear wave equations with shock wave solutions.<br \/>\nNonlinear wave equations with solitary wave solutions.<br \/>\nInitial value p<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32703\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\">\u7b2c\u4e09\u9031<\/td><td class=\"column-2\">Classification of 3 types of second order linear PDEs (I).<br \/>\nInitial value problem for a whole-line linear wave equation and the dAlembert solutions (II).<br \/>\nInitial-boundary value problem for a half-line linear wave equation.<br \/>\nInitial-boundary value problem fo<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32709\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\">\u7b2c\u56db\u9031<\/td><td class=\"column-2\">Initial-boundary value problem for a finite-line linear wave equation (II).<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32715\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\">\u7b2c\u4e94\u9031<\/td><td class=\"column-2\">Linear superposition and sub-problems.<br \/>\nMethod of Separation of Variables.<br \/>\nFourier series representations of solutions.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32721\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\">\u7b2c\u516d\u9031<\/td><td class=\"column-2\">Classification of 3 types of second order linear PDEs (II).<br \/>\nInitial value problem for a whole-line linear heat equation solved by the Fundamental solution.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32727\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\">\u7b2c\u4e03\u9031<\/td><td class=\"column-2\">Initial-boundary value problem for a finite-line linear heat equations solved by method of Separation of Variables.<br \/>\nInitial value problem for an infinite-line linear heat equation solved by Fourier transform and inverse Fourier transform.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32733\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\">\u7b2c\u516b\u9031<\/td><td class=\"column-2\">Boundary value problem for a Laplace\u2019s equation in a rectangle solved by method of Separation of Variables.<br \/>\nBoundary value problem for a Laplace\u2019s equation in a circle solved by method of Separation of Variables.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32739\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\">\u7b2c\u4e5d\u9031<\/td><td class=\"column-2\">Boundary value problem for a Poisson\u2019s equation in a circle.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32748\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\">\u7b2c\u5341\u9031<\/td><td class=\"column-2\">Well-posed problems for linear PDE systems (I).<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32756\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\">\u7b2c\u5341\u4e00\u9031<\/td><td class=\"column-2\">Well-posed problems for linear PDE systems (II).<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32767\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\">\u7b2c\u5341\u4e8c\u9031<\/td><td class=\"column-2\">Well-posed problems for linear PDE systems (III).<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32773\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\">\u7b2c\u5341\u4e09\u9031<\/td><td class=\"column-2\">Nonlinear problems (I) -<br \/>\nThe effect of a combination of nonlinearity and dispersion;<br \/>\nThe effect of a combination of nonlinearity and dissipation;<br \/>\nThe effect of a combination of nonlinearity, dispersion, and dissipation.<br \/>\n<br \/>\nShock waves, steady-state solution<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32781\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\">\u7b2c\u5341\u56db\u9031<\/td><td class=\"column-2\">Nonlinear problems (II) - kdV equation and the solitary solutions.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32787\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\">\u7b2c\u5341\u4e94\u9031<\/td><td class=\"column-2\">Nonlinear Problems (III) - : Three famous universal nonlinear PDEs - kdV, s-G, and NLS equations.<br \/>\nCompletely integrable systems.<br \/>\ns-G equation and the travelling wave solutions.<br \/>\nNLS equation and the solitary wave solutions.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32793\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\">\u7b2c\u5341\u516d\u9031<\/td><td class=\"column-2\">Nonlinear Problems (IV) - Introduction of Riemann surfaces of genus N (1) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32799\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\">\u7b2c\u5341\u4e03\u9031<\/td><td class=\"column-2\">Nonlinear Problems (V) - Introduction of Riemann surfaces of genus N (2) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS.<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32805\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-66 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-5f099062\" data-id=\"5f099062\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5b79cc32 elementor-widget-divider--view-line_text elementor-widget-divider--element-align-center elementor-widget elementor-widget-divider\" 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Prob&hellip;<\/p>\n","protected":false},"featured_media":0,"parent":29373,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","format":"standard","meta":{"footnotes":""},"categories":[1405],"tags":[],"class_list":["post-32793","course_page","type-course_page","status-publish","format-standard","hentry","category-introduction-to-partial-differential-equations-103"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Nonlinear Problems (III) - : Three famous universal nonlinear PDEs - kdV, s-G, and NLS equations. 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