This paper rigorously establishes the existence and number of periodic wave solutions for a perturbed generalized KdV (pgKdV) equation with two high-order nonlinearities <Equation ID="Equ56"> <EquationSource Format="TEX">$\begin{aligned} (u^m)_t + (u^{m+\ell })_x + u_{xxx} + \varepsilon (u_{xx} + u_{xxxx}) = 0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msub> <mo>+</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mrow> <mi>m</mi> <mo>+</mo> <mi>ℓ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xxx</mi> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xxxx</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where m and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> are arbitrary positive integers, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter. By combining geometric singular perturbation theory with the analysis of Abelian integral, we give the condition of wave speed for the existence and number of periodic wave solutions. Furthermore, the monotonicity of the limit wave speed is obtained by a classical and effective mathematical analysis method. Additionally, the upper and lower bounds of limit wave speed are derived. To our knowledge, this is the first complete result concerning the existence and number of periodic wave solutions for this class of pgKdV equation with arbitrary positive integers m and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>.

Periodic Waves and Monotonicity of Limit Wave Speed in a Perturbed Generalized KdV Equation with Two Arbitrarily High-Order Nonlinearities

期刊论文
发表日期：

2026年5月20日

查看全文

Xiuli Cen,
Chunhang Zhang

摘要

This paper rigorously establishes the existence and number of periodic wave solutions for a perturbed generalized KdV (pgKdV) equation with two high-order nonlinearities $\begin{aligned} (u^m)_t + (u^{m+\ell })_x + u_{xxx} + \varepsilon (u_{xx} + u_{xxxx}) = 0, \end{aligned}$ $\begin{matrix} {(u^{m})}_{t} + {(u^{m + ℓ})}_{x} + u_{xxx} + ε (u_{xx} + u_{xxxx}) = 0, \end{matrix}$ where m and $\ell $ $ℓ$ are arbitrary positive integers, and $\varepsilon >0$ $ε > 0$ is a small parameter. By combining geometric singular perturbation theory with the analysis of Abelian integral, we give the condition of wave speed for the existence and number of periodic wave solutions. Furthermore, the monotonicity of the limit wave speed is obtained by a classical and effective mathematical analysis method. Additionally, the upper and lower bounds of limit wave speed are derived. To our knowledge, this is the first complete result concerning the existence and number of periodic wave solutions for this class of pgKdV equation with arbitrary positive integers m and $\ell $ $ℓ$ .