Camassa–Holm–KP Equation Analysis

• The CHKP equation is a nonlinear dispersive model that unifies high-order x-dispersion with 2D transverse modulation, accurately modeling rogue waves in various physical contexts.
• Symbolic computation and Hirota-type bilinear methods enable the construction of exact multi-rogue wave solutions with controlled lump positions and interactions.
• Advanced HDG discretizations provide energy-stable and high-fidelity numerical approximations, essential for simulating both regular and singular wave phenomena.

The Camassa–Holm–Kadomtsev–Petviashvili (CHKP) equation represents a class of nonlinear dispersive partial differential equations that unify high-order $x$-dispersion effects (as found in Camassa–Holm type models) with two-dimensional transverse modulation (Kadomtsev–Petviashvili dynamics). These equations arise in contexts such as shallow water wave modeling and plasma dynamics, where both nonlinearity and multidimensional dispersion are critical for capturing localized, transient phenomena such as solitary waves and rogue wave clustering. This entry presents an integrated exposition of the two-dimensional CHKP equation, its generalized extension (gCHKP), exact multi-rogue wave solutions via symbolic computation, and advanced HDG discretizations enabling efficient and stable numerical approximation.

1. Mathematical Formulation of the CHKP and gCHKP Equations

The foundational two-dimensional CH–KP equation for a scalar wave-profile $u(x,y,t)$ is given by

$$\bigl(u_{t}-u_{t xx}+2\kappa u_{x}+3u\,u_{x} -2u_{x}u_{x x}-u\,u_{x x x}\bigr)_{x} +u_{yy} =0, \qquad (x,y)\in\mathbb R^2,\;t>0,$$

where $\kappa >0$ denotes the background depth parameter. An equivalent nonlocal evolution form is

$$u_t +(1-\partial_x^2)^{-1} \Bigl[ f(u)_x -2u_xu_{xx} -u\,u_{xxx} +\partial_x^{-1}u_{yy} \Bigr] =0,$$

with $f(u)=2\kappa u+\frac{3}{2}u^2$, and the nonlocal term $\partial_x^{-1}u_{yy}$ encapsulating nontrivial transverse effects (Dwivedi et al., 20 Jan 2026).

The generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCHKP) equation extends this structure: $$\bigl(u_t + \alpha\,u_x + \beta\,u\,u_x + \Upsilon\,u_{xxt}\bigr)_x + u_{yy} = 0,$$ where $\alpha$, $\beta$, and $\Upsilon$ respectively denote linear advection, nonlinear advection, and high-order CH-type dispersion coefficients. In the limit $\partial_y\to0$, the gCHKP reduces to an integrable Camassa–Holm equation; for vanishing high-order dispersion the KP regime is recovered (Liu et al., 2019).

2. Symbolic Computation and Bilinearization Methods

Exact multi-rogue wave solutions to the gCHKP equation are constructed through a symbolic computation approach, leveraging a Hirota-type bilinear transformation. Upon the parameter constraint $\beta=6\Upsilon$, one posits

$$u(x,y,t) = \tau_0 + 2\,\frac{\partial^2}{\partial\sigma^2}\ln\,\xi(\sigma,y),\quad\sigma=x+t,$$

and derives a single polynomial bilinear equation for $\xi(\sigma,y)$. The solution methodology requires:

• Ansatz selection for $\xi$ as a polynomial in $\sigma$ and $y$, with degree matching the desired rogue wave count.
• Substitution into the bilinear equation, producing an algebraic system where coefficients of all independent monomials are set to zero.
• Algebraic solution for all undetermined polynomial coefficients, performed via symbolic software (e.g., Mathematica, Maple), absent determinant representations.
• Reconstruction of the wave field $u$ by differentiation and logarithmic transformation (Liu et al., 2019).

3. Explicit Multi-Rogue Wave Solutions

Closed-form rogue wave solutions for the gCHKP are provided for one, three, and six lumps:

• One-rogue wave (degree 2 polynomial):

$$\xi(\sigma,y) = (\sigma-\mu)^2 + \theta\,(y-\nu)^2 + \vartheta_0,\quad \vartheta_0 = -\frac{3\,\Upsilon}{\theta},\quad \theta = \alpha + 6\,\tau_0\,\Upsilon+1$$

$$u(\sigma,y) = \tau_0 + \frac{4\left[-\dfrac{3\Upsilon}{\theta}-(\mu-\sigma)^2+\theta\,(y-\nu)^2\right]}{\left[-\dfrac{3\Upsilon}{\theta}+(\mu-\sigma)^2+\theta\,(y-\nu)^2\right]^{2}}$$

• Three-rogue wave (degree 6 polynomial):

$$u(\sigma,y) = \tau_0 + 2\,\Bigl\{ \frac{\xi_{\sigma\sigma}}{\xi} - \frac{(\xi_\sigma)^2}{\xi^2} \Bigr\}$$

with $\xi$ of prescribed degree and structure, the two free parameters ($\theta_{19}$, $\theta_{22}$) governing lump positions and inter-distance.

• Six-rogue wave (degree 12 polynomial):

Analogous closed form with $\sim40$ polynomial coefficients, all determined via symbolic matching.

For all cases, the amplitude at each lump exceeds twice the background $\tau_0$, confirming the “rogue” nature. Lump positions and interactions are parameter-controlled (Liu et al., 2019).

4. Wave Dynamics, Localization, and Physical Interpretation

Wave patterns generated by gCHKP multi-lump solutions display critical-point structures and localization properties governed by dispersion and transverse terms:

• The one-lump case reveals a central dip and four surrounding symmetric peaks. For negative $\Upsilon$ a sharp crest is observed, while positive $\Upsilon$ yields a reversed “dip-in-crest” profile.
• Three lumps manifest as triangular configurations with non-fusing, nonlinear interactions under contour and 3D surface examination.
• Six-lump solutions form rings of localized peaks with reflection symmetry in $x$, $y$, and sharply concentrated energy density.

The two-dimensional term $u_{yy}$ ensures true multidimensional rogue localization, and Camassa–Holm dispersion $u_{xxt}$ leads to sharper, taller waveforms along the primary direction (Liu et al., 2019).

5. Advanced Spatial Discretization: Hybridizable Discontinuous Galerkin Methods

Numerical resolution of CHKP-type equations is enabled by hybridizable discontinuous Galerkin (HDG) schemes, designed to localize nonlocal operators and achieve energy-stable approximations. The scheme includes:

• Nonlocal Term Localization: Introduction of an auxiliary variable $v$ with $v_x=u_y$, converting $\partial_x^{-1}u_{yy}$ into a local system amenable to elementwise computation.
• HDG Structure: Use of tensor-product polynomial spaces $Q_k(I_i)\otimes P_k(J_j)$ on Cartesian meshes, with broken bulk and skeleton trace spaces, enabling efficient differential operator treatment across mesh elements.
• Energy Stability: The method achieves unconditional $L^2$ stability through carefully chosen numerical fluxes and stabilization parameters, as established by a discrete energy identity.
• Convergence: Theoretical error bounds ensure $\mathcal O(h^{k+1/2})$ convergence for solution and derivative variables under regularity assumptions ($u\in C([0,T];H^{k+3})$), with an explicit error inequality derived via projection decomposition and standard analysis (Dwivedi et al., 20 Jan 2026).

6. Computational Results and Practical Capabilities

Numerical tests validate the HDG framework:

• For smooth manufactured solutions (e.g., $u(x,y,t)=e^{-t}\sin x\sin y$), order-optimal convergence rates $O(h^{k+1})$ are observed up to $k=3$, with explicit error data on uniform Cartesian grids.
• For strongly nonlinear peakons ($u(x,y,t)=c\,\exp(-|x+y-ct|)$), the HDG scheme captures sharp solitary wave crests with minimal numerical dissipation or Gibbs-type oscillations, maintaining phase accuracy below $10^{-3}$ over extended propagation.
• The method resolves both regular and singular waveforms with high fidelity and mesh independence, demonstrating rapid convergence and robust sharp wave propagation across time (Dwivedi et al., 20 Jan 2026).

7. Research Implications and Future Directions

The unification of Camassa–Holm and KP physics in the multidimensional CHKP framework enables analytic and computational exploration of clustered and interacting rogue waves in integrable systems. Symbolic-computation-based solution construction, as demonstrated for gCHKP, offers an extensible platform for analytic studies with controllable lump positions and numbers, with algorithmic scalability subject to algebraic complexity constraints. Physically, this facilitates modeling of isolated energy events in shallow-water, plasma, and nonlinear optical systems.

Continued research directions include extending the methods to variable coefficient regimes (e.g., inhomogeneous media), coupling with shear flows, and further refinement of numerical schemes for more general boundary conditions or irregular geometries. The analytical and numerical integration of nonlocal and transverse effects remains central to accurate simulation and theoretical characterization of multidimensional dispersive wave phenomena (Liu et al., 2019, Dwivedi et al., 20 Jan 2026).