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Peakons and pseudo-peakons of higher order b-family equations

Si-Yu Zhu, Ruo-Xia Yao, De-Xing Kong, S. Y. Lou

TL;DR

This work extends the theory of peakons and pseudo-peakons to higher-order b-family (J-bF) equations by proposing and numerically verifying (via MAPLE) three central conjectures: a $b$-independent pseudo-peakon, a $b$-independent peakon, and a $b$-dependent peakon, across increasing orders of J. For specific cases with $J=2,3,4,5$, the authors derive explicit weak-solution ansatzes and show the emergence of 3rd-order pseudo-peakons and higher-order pseudo-peakons under parameter constraints, along with both $b$-independent and $b$-dependent peakons, highlighting intricate interactions between the parameters $b$ and $J$. The results generalize known lower-order peakon behavior (CH and DP) and reveal a layered structure where pseudo-peakons can shift to higher-order forms under constraints, while peakons manifest in $b$-independent and $b$-dependent families. The study lays groundwork for rigorous proofs, stability analyses, and potential physical applications in nonlinear wave dynamics.

Abstract

This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order $b$-family equations, referred to as the $J$-th $b$-family ($J$-bF) equations. We propose several conjectures concerning the weak solutions of these equations, including a $b$-independent pseudo-peakon solution, a $b$-independent peakon solution, and a $b$-dependent peakon solution. These conjectures are analytically verified for $J \leq 14$ and/or $J \leq 9$ using the computer algebra software MAPLE. The $b$-independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants, with higher-order pseudo-peakons derived under specific parameter constraints. Additionally, we identify both $b$-independent and $b$-dependent peakon solutions, highlighting their distinct properties and the nuanced relationship between the parameters $b$ and $J$. The existence of these solutions underscores the rich dynamical structure of the $J$-bF equations and generalizes previous results for lower-order equations. Future research directions include higher-order generalizations, rigorous proofs of the conjectures, interactions between different types of peakons and pseudo-peakons, stability analysis, and potential physical applications. These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.

Peakons and pseudo-peakons of higher order b-family equations

TL;DR

This work extends the theory of peakons and pseudo-peakons to higher-order b-family (J-bF) equations by proposing and numerically verifying (via MAPLE) three central conjectures: a -independent pseudo-peakon, a -independent peakon, and a -dependent peakon, across increasing orders of J. For specific cases with , the authors derive explicit weak-solution ansatzes and show the emergence of 3rd-order pseudo-peakons and higher-order pseudo-peakons under parameter constraints, along with both -independent and -dependent peakons, highlighting intricate interactions between the parameters and . The results generalize known lower-order peakon behavior (CH and DP) and reveal a layered structure where pseudo-peakons can shift to higher-order forms under constraints, while peakons manifest in -independent and -dependent families. The study lays groundwork for rigorous proofs, stability analyses, and potential physical applications in nonlinear wave dynamics.

Abstract

This paper explores the rich structure of peakon and pseudo-peakon solutions for a class of higher-order -family equations, referred to as the -th -family (-bF) equations. We propose several conjectures concerning the weak solutions of these equations, including a -independent pseudo-peakon solution, a -independent peakon solution, and a -dependent peakon solution. These conjectures are analytically verified for and/or using the computer algebra software MAPLE. The -independent pseudo-peakon solution is a 3rd-order pseudo-peakon for general arbitrary constants, with higher-order pseudo-peakons derived under specific parameter constraints. Additionally, we identify both -independent and -dependent peakon solutions, highlighting their distinct properties and the nuanced relationship between the parameters and . The existence of these solutions underscores the rich dynamical structure of the -bF equations and generalizes previous results for lower-order equations. Future research directions include higher-order generalizations, rigorous proofs of the conjectures, interactions between different types of peakons and pseudo-peakons, stability analysis, and potential physical applications. These advancements significantly contribute to the understanding of peakon systems and their broader implications in mathematics and physics.

Paper Structure

This paper contains 7 sections, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Conjectures
  3. Fifth order $b$ family and its peakon and pseudo-peakon solutions
  4. Seventh order $b$ family and 3rd and 5th order pseudo-peakons
  5. Ninth order $b$ family and $3$rd, $5$th, and $7$th order pseudo-peakons
  6. Eleventh order $b$-family and 3rd, 5th, 7th, and 9th order pseudo-peakons
  7. Summary and discussions

Figures (4)

  • Figure 1: (a) Pseudo-peakon structure with one smooth peak and double bottoms expressed by \ref{['J3pp']} and $a = -2$; (b) W-shaped pseudo-peakon structure given by \ref{['J3pp1']} and $a = -50$; (c) M-shaped pseudo-peakon structure of \ref{['J3pp1']} with $a = 2$; (d) Pseudo-peakon structure with a single smooth peak expressed by \ref{['J3pp1']} and $a = \frac{1}{2}$; (e) The structure of the second-order derivative of the 3rd-order pseudo-peakon \ref{['J3pp1']} with $a = \frac{1}{2}$; (f) The structure of the fourth-order derivative of the 5th-order pseudo-peakon \ref{['J3pp1']} with $a = \frac{1}{3}$.
  • Figure 2: The graphs of peakons. (a) The structure of the $b$-independent peakon expressed by \ref{['J3p']}; (b) The structure of the $b$-dependent peakon of \ref{['J3bp']} with $b = 3$.
  • Figure 3: (a) A quite smooth pseudo-peakon \ref{['J4pp']} with $\{a_1 = \frac{1}{2},\ a_2 = \frac{1}{6}\}$ and $v_x|_{x=ct} = v_{xx}|_{x=ct} = v_{xxx}|_{x=ct} = 0$; (b) M-shaped pseudo-peakon \ref{['J4pp']} with $a_1 = a_2 = 1$; (c) The structure of the pseudo-peakon with three smooth peaks expressed by \ref{['J4pp']}, $a_1 = -\frac{5}{4}$ and $a_2 = 1$; (d) W-shaped pseudo-peakon \ref{['J4pp']} with $\{a_1 = 1,\ a_2 = -30\}$; (e) The structure of the fourth order derivative of $v$ with respect to $x$ for the 5th-order pseudo-peakon expressed by \ref{['J4pp']} with the same parameters as (a); (f) The structure of the sixth order derivative of $v$ with respect to $x$ for the 7th-order pseudo-peakon given by \ref{['J4pp']} with the selections $\{a_1 = \frac{2}{5},\ a_2 = \frac{1}{15}\}$.
  • Figure 4: (a) Pseudo-peakon with four smooth peaks expressed by \ref{['J5pp']} and $\{a_1 = 10,\ a_2 = -7,\ a_3 = 1\}$; (b) Pseudo-peakon with three smooth peaks given by \ref{['J5pp']} and $\{a_1 = 1,\ a_2 = -5,\ a_3 = 1\}$; (c) M-shaped pseudo-peakon expressed by \ref{['J5pp']} with $\{a_1 = 10,\ a_2 = 1,\ a_3 = 1\}$; (d) Pseudo-peakon with double smooth peaks expressed by \ref{['J5pp']} and $\{a_1 = \frac{1}{2},\ a_2 = \frac{1}{5},\ a_3 = \frac{1}{10}\}$; (e) Single smooth peaked pseudo-peakon expressed by \ref{['J5pp']} with $\{a_1 = \frac{3}{7},\ a_2 = \frac{2}{21},\ a_3 = \frac{1}{105}\}$; (f) W-shaped pseudo-peakon expressed by \ref{['J5pp']} with $\{a_1 = \frac{1}{2},\ a_2 = \frac{1}{5},\ a_3 = -2\}$; (g) The seventh-order derivative of $v$ with respect to $x$ of (e); (h) The eighth-order derivative of $v$ with respect to $x$ of (e).