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Systems of partial differential equations describing pseudo-spherical or spherical surfaces

Mingyue Guo, Jing Kang, Zhenhua Shi

Abstract

In this paper, we study systems of nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of systems of Camassa-Holm-type equations of the form \begin{equation*} \left\{ \begin{aligned} u_{t} - u_{xxt} &= F(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \\ v_{t} - v_{xxt} &= G(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \end{aligned} \right. \end{equation*} with $m,n\geq2$, for $F$ and $G$ smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of the results provide new examples of such systems, such as the Song-Qu-Qiao system, the two-component Camassa-Holm system with cubic nonlinearity, and the modified Camass-Holm-type system. Moreover, we construct the nonlocal symmetry and a non-trivial solutions for the two-component Camassa-Holm system with cubic nonlinearity from the gradients of spectral parameters.

Systems of partial differential equations describing pseudo-spherical or spherical surfaces

Abstract

In this paper, we study systems of nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of systems of Camassa-Holm-type equations of the form \begin{equation*} \left\{ \begin{aligned} u_{t} - u_{xxt} &= F(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \\ v_{t} - v_{xxt} &= G(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \end{aligned} \right. \end{equation*} with , for and smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of the results provide new examples of such systems, such as the Song-Qu-Qiao system, the two-component Camassa-Holm system with cubic nonlinearity, and the modified Camass-Holm-type system. Moreover, we construct the nonlocal symmetry and a non-trivial solutions for the two-component Camassa-Holm system with cubic nonlinearity from the gradients of spectral parameters.
Paper Structure (7 sections, 10 theorems, 133 equations)

This paper contains 7 sections, 10 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results and Examples
  4. Classification Theorems
  5. Examples
  6. Nonlocal symmetry
  7. Concluding and remarks

Key Result

Lemma 3.1

The necessary and sufficient conditions for a system of partial differential equations 3.1 to describe a pseudospherical surface ($\delta = 1$) or spherical surface ($\delta = -1$), with associated functions $f_{ij}=f_{ij}(x,t,u,u_1,\dots,u_m,v,v_1,\dots,v_n)$, are given by

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Proof
  • Corollary 3.2
  • Corollary 3.3
  • Proof
  • Theorem 3.4
  • Proof
  • Theorem 3.5
  • ...and 16 more